The Beilinson-Bernstein Localization Theorem for
the affine Grassmannian
MAASSACHUSETTS Iw
OF TECHNOLOGY
by
Giorgia Fortuna
JUL 2 5 2013
Laurea in Matematica,
Universita' di Roma "La Sapienza" (2008)
B;RZZARIES
Submitted to the Department of Mathematics
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2013
@ Massachusetts Institute of Technology 2013. All rights reserved.
i2/
A uthor .......................
............
. ..
.
...
.
..
... ..
Department of Matheatc
/
May 21, 2013
Certified by ..................
Dennis Gaitsgory
Professor of Mathematics
Thesis Supervisor
A ccepted by .........................................................
Paul Seidel
Chairman, Department Committee on Graduate Theses
Ey
2
The Beilinson-Bernstein Localization Theorem for the affine
Grassmannian
by
Giorgia Fortuna
Submitted to the Department of Mathematics
on May 21, 2013, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Abstract
This thesis mainly deals with the study of the category 'cit-mod of admissible
modules for the affine Kac-Moody algebra at the critical level 'cit and the study of
its center Z(, it-mod). The language used in this work is the one of chiral algebras,
viewed as either D-modules over a smooth curve X or as a collection of sheaves
of the curve. In particular, we study the chiral algebra Ait
over powers X(
corresponding to the Kac-Moody algebra e,.it and its center 3cit. By considering
the categories of Acrit-modules and 3crit-modules supported at some point x in X,
we recover the categories e,.it-mod and Z(r.it-mod)-mod respectively. In this thesis
we also study the chiral algebra Dit of critically twisted differential operators on
the loop group G((t)) and its relation to the category of D-modules over the affine
Grassmannian GrG,x = G((t))/G[[t]].
In the first part of the thesis, we consider the chiral algebra Ait and its center
3,it. The commutative chiral algebra 3,it admits a canonical deformation into
a non-commutative chiral algebra Wh. We will express the resulting first order
deformation via the chiral algebra Dit of chiral differential operators on G((t)) at
the critical level.
In the second part of the thesis, we consider the Beilinson-Drinfeld Grassmannian
GrG and the factorization category of D-modules on it. We try to describe this category in algebraic terms. For this, we first express this category as the factorization
category 'Deit-mod JG of chiral 'Derit-modules which are equivariant with respect to
the action of a certain factorization group JG. Then we express the factorization
category of chiral 3erit-modules as the category of modules over the factorization
space Op' of opers on the punctured disc. Using the Drinfeld-Sokolov reduction 9,
we construct a chiral algebra '3 and a functor Iq, from the category of D-modules on
GrG to the category of chiral '3-modules that are supported on a certain sub-scheme
of Op'. We conjecture that this functor establishes an equivalence between these
3
two categories.
Thesis Supervisor: Dennis Gaitsgory
Title: Professor of Mathematics
4
Acknowledgments
First of all, I would like to thank my advisor Dennis Gaitsgory without whom I
would have been lost an endless number of times. He has always been a source of
enthusiasm and ideas during my five years at MIT. I would like to thank him for his
patience and for teaching me how to see things always "one categorical level up",
whatever that means. I would also like to thank him for the enormous number of
suggestions he gave me, and for making me realize that being wrong about something
is the best way of learning.
I would also like to thank Sam Raskin, Dario Beraldo and Sasha Tsymbaliuk for
helping me getting through these years at MIT. In particular I would like to thank
Sam for being always helpful and willing to translate the"Gaitsgorian" into English
and Dario for some useful discussions we had in the last year. I would like to thank
Sasha for his endless support, even in the moments where he had no idea he was
helping me.
A special thanks goes to many professors at MIT that taught me how different
math and mathematicians can be, and gave me the possibility of learning what I
would have never learned in any other place.
A special thanks goes to Michael
Artin for teaching me Italian, and to Pavel Etingof, for being the living example
of a person that knows everything about math. I would also like to thank Roman,
Gigliola and Paul for their nice words and their support during this past month (I
am not referring to the bombing).
Another special thanks goes to my dying computer, or I should probably say to
Skipe, that is killing the two of us, and whoever happens to be on the other side
of the screen. A special thanks goes to Giacomo and Gabriele who got the worse
of me but tried to be as supportive as possible, especially Giacomo who still stands
me even after a year of talking about bed bugs and a year of blaming him for what
we could have done differently and we haven't.
5
Lastly, and most importantly, I would like to thank the GAS production and
all the people that worked for it. In particular a special thanks goest to Andrea
Appel and Salvatore Stella .... they just deserve it, I am sure that living with me has
not been easy. A special thanks to Roberto Svaldi as well, for his way of cheering
you up with the most obscene sentences he could come up with. Another special
thanks to Luis for being always nice to me, even when all I wanted to do was to kill
him. To finish, a special thanks to Luca Belli and Jeremy Russell, because without
them this year would have been much worse, but most importantly, without them,
I would have never realized that the Queen is the best, and that "you can be happy
because you know why you are sad, which is because you have not been nice enough
to the bird."
6
Contents
1
1.1
W-algebras and chiral differential operators on the loop group
1.1.1
1.2
1.3
2
11
Introduction
Main Theorem
.
.
.
. . . . . . . . . . . . . . . . . . . . . . . . . . 15
D-modules over the affine Grassmannian
. . . . . . . . . . . . . . . .
The Beilinson-Bernestein Localization Theorem
1.2.2
Construction of the functor
1.2.3
The factorization picture . . . . . . . . . . . . . . . . . . . . . 21
1.2.4
The main conjecture
1.2.5
How conjecture 1.2.3 implies conjecture 1.2.2 . . . . . . . . . . 25
Organization of the thesis
. . . . . . . .
2.2
17
1.2.1
Chiral algebras
17
. . . . . . . . . . . . . . . . . . . 20
. . . . . . . . . . . . . . . . . . . . . . . 24
. . . . . . . . . . . . . . . . . . . . . . . .
W-algebras and chiral differential operators at the critical level
2.1
13
26
31
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.1.1
Commutative chiral algebras . . . . . . . . . . . . . . . . . . .
33
2.1.2
Chiral envelope of Lie*-algebras . . . . . . . . . . . . . . . . .
35
2.1.3
The chiral algebra A . . . . . . . . . . . . . . . . . . . . . . ..
36
2.1.4
Lie*-algebroids and '-extensions
. . . . . . . . . . . . . . . . 37
Quantization-deformation of commutative chiral algebras . . . . . . . 41
2.2.1
Quantizations of chiral-Poisson algebras
Quantization of the center
2.2.2
. . . . . . . . . . . . 42
3,t . . . . . . . . . . . . . . . . . 43
Construction of the inverse . . . . . . . . . . . . . . . . . . . . 43
7
Definition of Indh(Lc)
. . . . . . . . .
45
. .. . . . . . . .
46
Construction of IndT(L). . . . . . . . .
48
Different construction of Indh (LC).
50
A special case: deformations of 'R.
52
The construction of c('R). . . . . . . .
52
The classical setting
2.3
Quantization of the center
. . . . . . . .
59
The Drinfeld-Sokolov reduction . . . .
60
The BRST reduction . . . . . . . . . .
60
BRST reduction for modules . . . . . .
62
Drinfeld-Sokolov reduction . . . . . . .
62
Main theorem . . . . . . . . . . . . . . . . . .
64
2.3.1
2.4
3
2.4.1
The chiral algebra of twisted differential operators on the loop
group
2.4.2
2.4.3
cit
. . . . . . . . . . . . . . . . . .
64
Proof of the theorem . . . . . . . . . .
67
Reformulation of the Theorem . . . . .
67
Construction of the map F . . . . . . .
72
Definition of the map
2.4.4
3
f.
. . . . . . . .
73
End of the proof of Theorem 2.6 . . . .
75
Proof of Proposition 2.4.1 . . . . . . .
77
Localization theorem for the affine Grassmannian
85
Factorization categories and factorization spaces . .
85
. . . . . . . . . . . . . . . .
86
3.1
3.1.1
The Ran space
3.1.2
Factorization categories
. . . . . . . . . . .
89
3.1.3
factorization algebras . . . . . . . . . . . . .
90
3.1.4
Chiral modules over X . . . . . . . . . . .
92
3.1.5
Factorization spaces
96
. . . . . . . . . . . . .
8
3.1.6
From 'Dx-schemes to factorization spaces . . . . . . . . . . . .
97
The factorization space attached to a 'Dx-scheme . . . . . . .
99
3.1.7
Factorization groups
. . . . . . . . . . . . . . . . . . . . . . . 100
3.1.8
Chiral modules over a commutative chiral algebra . . . . . . . 101
Chiral algebras and topological algebras attached to them
3.2
3.3
. .
103
Action of a group scheme on a category . . . . . . . . . . . . . . . . . 107
3.2.1
Strong action on C . . . . . . . . . . . . . . . . . . . . . . . . 108
3.2.2
The case C = A-mod. . . . . . . . . . . . . . . . . . . . . . . . 109
Action of a factorization group on a factorization category
. . . . . . 111
3.3.1
Action on the category A-mod . . . . . . . . . . . . . . . . . . 113
3.3.2
Strong action on the category A-mod . . . . . . . . . . . . . . 115
3.3.3
Strong action on the category 'Dc,it-mod . . . . . . . . . . . . 120
Action of Jx(G) on the chiral algebra A . . . . . . . . . . . ..
120
Strong action of Jx(G) on 'D,.it-mod . . . . . . . . . . . . . . 121
3.4
The Beilinson-Drinfeld Grassmannian and critically twisted D-modules
on it . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
3.4.1
The Beilinson-Drinfeld Grassmannian . . . . . . . . . . . . . . 123
3.4.2
Critically twisted D-modules on GrG . . . . . .
Twisted D-modules on GrG
. . . . . .
- - - - - -.
- - - - - - - - - - - - - 130
Critically twisted differential operators on GrG,I . . . . . . . .
3.4.3
127
132
4
Construction of the line bunde Z,.it
over the Beilinson-Drinfeld
Grassmannian . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
3.4.4
D-modules on the Beilinson-Drinfeld Grassmannian as chiral
'D ,.it-modules
. . . . . . . . . . . . . . . . . . . . . . . . . . . 138
The equivalence over the point . . . . . . . . . . . . . . . . . . 138
The equivalence over X
3.5
. . . . . . . . . . . . . . . . . . . . . 140
The space of Opers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
3.5.1
The space of opers
. . . . . . . . . . . . . . . . . . . . . . . . 145
9
3.5.2
Opers on the punctured disc . . . . . . . . . . . . . . . . . . . 147
Relation with the center
3.5.3
3.6
Unramified opers . . . . . . . . . . . . . . . . . . . . . . . . . 149
The Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
3.6.1
3.7
3,a . . . . . . . . . . . . . . . . . . 148
The conjecture over X . . . . . . . . . . . . . . . . . . . . . ..
150
Drinfeld-Sokolov reduction for modules over X .
. . . . . . ..
152
BRST-reduction for modules on X . . . . . . . . . . . . . . . .
152
The localization Conjecture for the Beilinson-Drinfeld Grassmannian
154
3.7.1
Definition of the functor 1r,
. . . . . . . . . . . . . . . . . . . 154
Appendices
159
A How to prove conjecture 3.6.1
161
A.1
First approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
A.1.1
Construction of the inverse to I,
A.2 Second approach
A.2.1
. . . . . . . . . . . . . . . . 162
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
The case G = T . . . . . . . . . . . . . . . . . . . . . . . . . . 167
10
Chapter 1
Introduction
Let g be a finite dimensional simple Lie algebra over an algebraically closed field
k of characteristic 0, and let G be the corresponding adjoint algebraic group. Fix
a smooth curve X over k, and a point x
any invariant bilinear form
tj,
E X. Let t be a coordinate near x. For
denote by g
the Kac-Moody algebra given as the
corresponding central extension of the loop algebra g 0 C((t)) by C. Denote by
the Killing form on g and let
,.it be the critical level
,.it =
-1/2nKill.
rKill
For the
critical level, denote by U'it the appropriately completed twisted enveloping algebra
of
cit.
Consider the category
the same as discrete admissible
eit-mod
of continuous modules over U'it. These are
eit-modules
on which 1
E C acts as the identity.
At the critical level, unlike any other r,, the center of the category
eit-mod
is non-
trivial. For instance, in the case of s12 , it is generated by the Sugawara operators.
We denote by
3
,.it the center
,.it := Z('e,-it-mod )
This peculiarity makes the theory of
eit-modules
more interesting and more com-
plicated.
Denote by g the Langlands dual Lie algebra to g, and let X be a smooth curve
11
over the complex numbers. Denote by Op ,x the space of b-opers on X, introduced
in [BD2].
Roughly speaking, an oper is a triple (To, 7,
bundle on X, 37
V), where To is a
O-
is a reduction to a fixed Borel b c 0 and V is a connection on
.To satisfying certain properties. For every point x E X, we denote by Op,(D ) the
ind-scheme of opers on the punctured disc D' = Spec(C((t))), and by Op§(Dx) the
scheme of regular opers i.e., opers on the disc D = Spec(C[[t]]).
The interplay between the representation theory of ' it and the space Op§,x
is given by a theorem of Feigin and Frenkel.
In [FF], they prove the existence
of an isomorphism of commutative topological algebras
5,it
~ Fun(Op (D*)).
In the work [FG2] of D. Gaitsgory and E. Frenkel, they define a closed sub-indscheme of Opj(D ), denoted by Opgm," corresponding to unramified opers. This
ind-scheme consists of those opers that are unramified as local systems. We denote
by ' it-modeg the subcategory of '
t-mod on which the action of Fun(Opb(D*))
factors through Fun(Opb(Dx)). Similarly we denote by 'git-modun, the subcategory consisting of modules on which the action of Fun(Op6(D*)) factors through
the quotient Fun (Op,n)
Our basic tool in this paper is the theory of chiral algebras as introduced in [BD].
In fact, we will see in 2.1.1 how we can attach to the 'Dx-algebra Opbx a commutative chiral algebra, and more generally, how an affine 'Dx-scheme corresponds to
a commutative chiral algebra. This suggests that the theory of chiral algebras is a
more suitable tool for the study of the above categories. In particular we will use
the chiral algebra A, attached to & as defined in [AG].
is a Dx-module A equipped with a map
{, }
A chiral algebra on X,
: j~j*(A Z A) -+ Ai(A), called the
chiral product, satisfying certain properties, where A : X -+ X 2 denotes the diagonal embedding and
j
the inclusion of the complement. We will denote by [,]A the
restriction of the chiral product to A Z A -4 j*j*(A 0 A). The rising interest in the
theory of chiral algebras has a twofold motivation. The first is that it has numerous
12
applications in the study of conformal field theory in two dimensions. The second
is that, as explained in [BD], chiral algebras are the same as factorization algebras
on Ran(X), i.e., a sequence of quasi-coherent sheaves A(n) for every power of the
curve X", satisfying certain factorization properties. For instance, a co-unital affine
factorization space on Ran(X) is the same as a commutative chiral algebra on X.
This description makes the understanding of factorization algebras, and of modules
over them, easier.
1.1
W-algebras and chiral differential operators
on the loop group
Consider the chiral algebra A, attached to &.
3
c,it
the center of Ait
:= A,..i.
property that the fiber ( 3
c,.it),
For r, =
,-it = -AKkill
denote by
This is a commutative chiral algebra with the
over any point x E X is equal to the commutative
algebra Endcri(Vc,.it), where
IndO cri C.
V,c,.it := Ind
,0
The chiral algebra
3,it is closely related to the center
3
of the category geit-mod.
eit
In fact, for any chiral algebra A and any point x E X, we can form an associative
topological algebra A
with the property that its discrete continuous modules are
the same as A-modules supported at x (see [BD] 3.6.6). In this case the topological
associative algebra corresponding to
3
cit
is isomorphic to
crit.
As we mentioned before, the importance of choosing the level r, to be scit relies
on the fact that the center
,it happens to be very big. Another crucial feature of the
critical level is that Lcrit carries a natural Poisson structure, obtained by considering
the one parameter deformation of
rit given by scrit
13
+
hIkiil,
as explained below
in the language of chiral algebras. Moreover, according to [FF, F1, F2], the center
3,it is isomorphic, as Poisson algebra, to the quantum Drinfeld-Sokolov reduction of
U'(s,.t) introduced in [FF] and [FKW]. In particular the above reduction provides
a quantization of the Poisson algebra
,it that will be central in this work.
Since the language we have chosen is the one of chiral algebras, we will now
reformulate these properties for the algebra
The commutative chiral algebra
3
,it.
3,it can be equipped with a Poisson structure
which can be described in either of the following two equivalent ways:
" For any h
$
0 let r, be any non critical level
Ah the chiral algebra A,.
K =
c,-it
+ hkik
and denote by
Let z and u be elements of 3,.it. Let zK and w" be
any two families of elements in As such that z = zK and w = wK when h = 0.
Define the Poisson bracket of z and w to be
{z, W} =
[zK,
WK]A,
h
(mod h).
" The functor Tx of semi-infinite cohomology introduced in [FF] (which is the
analogous of the quantum Drinfeld-Sokolov reduction mentioned before and
whose main properties will be recalled later), produces a 1-parameter family of
chiral algebras {Wh} := {'x(Ah)} such that Wo ~- 3,it. Define the Poisson
structure on
3
cit
as
{z, W} = [hIih
(mod h)
where z = zal,_o and w = iGalao.
Although the above two expressions look the same, we'd like to stress the fact that,
unlike the second construction, in the first we are not given any deformation of 3,it.
In other words the elements zK and w, do not belong to the center of A, (that in
14
fact is trivial).
It is worth noticing that the associative topological algebras Ih
associated to them (usually denoted by Wh) are the well known W-algebras.
As in the case of usual algebras, the Poisson structure on
Kshler differentials Q£( 3 ,it)
3,it gives the sheaf of
a structure of Lie* algebroid. A remarkable feature,
when dealing with chiral algebras, is that the existence of a quantization {Wh} of
3
allows us to construct what is called a chiral extension Qc(
cit
3
,it)
of the Lie*
algebroid Q'(3,.it), and moreover, as it is explained in [BD] 3.9.11, this establishes
an equivalence of categories between 1-st order quantizations of
tensions of W(3c,.
1.1.1
3,it and chiral ex-
This equivalence is the point of departure for this work.
Main Theorem
In [BD], the highly non-trivial notion of chiral extension of Lie*-algebroid is introduced. Chiral extensions form a gerbe over a certain Picard category; in particular,
such extensions may not even exist. Given a chiral extension 2c of a Lie*-algebroid
, we can form its chiral envelope
U(2c)ch.
For example, for a Dx-space Y, and a
chiral extension of the Lie*-algebroid Oy of vector fields on Y, its chiral envelope is
a chiral algebra of twisted chiral differential operators on Y.
This project consists of comparing two, a priori different, chiral extensions of the
Lie*-algebroid Q1(3,). The first extension is defined using quantum W-algebras,
as explained before. Namely we consider the 1-paramenter family of chiral algebras
{Wa} :{=
Px(Ah)}. As is shown in [FF], the cohomology of Tx(A,-it) = XFx(Ao)
is concentrated in degree zero, and its 0-th cohomology is isomorphic to the center
3,it.
of
Therefore the chiral algebras {Wh} provide a 1-parameter family deformation
3,it, giving rise to the same Poisson structure as introduced above. According
to [BD], such a quantization gives rise to a chiral extension Qc(3c,-it) of Q'(3c-it).
The second extension is given via its chiral envelope.
We start with the chiral algebra Dit of critically-twisted differential operators
15
on the loop group G((t)), introduced in [AG]. The algebra 'Drit admits two embeddings of A,it, corresponding to right and left invariant vector fields on G((t)). The
two embedding 1 and r of Acrit into D,it endow the fiber ('Dit), with a structure
of
The fiber can therefore be decomposed according to these actions
eit-bimodule.
as explained below.
Consider the topological commutative algebra 3cit. For a dominant weight A,
let VA be the finite dimensional irreducible representation of g with highest weight
A and let 'V,
be the
e
it-module given by
IVA
Glerit
The action of the center
crit
U(erit)
®
VA.
U(g([t]jeC)
on V,,it factors as follows
$crit
End(V~cri).
-*->3c.,it A:=E
d(
c,1 )
Denote by IA the kernel of the above map, and consider the formal neighborhood
of Spec(3A.)
inside Spec(Ocrii). Let
be the full subcategory of
-it-modG[[t]]
c
modules such that the action of g[[t]] can be integrated to an action of G[[t]]. We
have the following Lemma.
Lemma 1.1.1. Any module M in 'eit-modG[[t|| can be decomposed into a direct
sum of submodules M such that each MA admits a filtration whose subquotients are
annihilated by A.
As a bimodule over 'cit the fiber at any point x E X of 'Dit is G[[t]] integrable with respect to both actions, hence we have two direct sum decompositions
of ('Dit), corresponding to the left and right action of 'cit, as explained in [FG2]:
('Drit)2= E
('Drit)\,
Adaninant
16
where ('Dit)A is the direct summand supported on the formal completion of Spec(3A it).
Denote by 'D,
the Dx-module corresponding to (Dcrit).
It is easy to see that Do
is in fact a chiral algebra.
Since the fiber of Acrit at x is isomorphic to V,
the embeddings 1 and r must
land in the chiral algebra 'DOe,.t Hence we have
The above two embeddings give 'DOit a structure of Acit-bimodule, hence it makes
sense to apply the functor of semi-infinite cohomology Tx to it with respect to both
actions. Let us denote by Cc-it the resulting chiral algebra
Co.,.
:= (x
0 Tx)(Do..t).
The main result of this work is the following.
Theorem 1.1. The chiral envelope U(Qc(3crit)) of the extension
0 3 3crit --
Qc(3crit) -+
Q(3crit) + 0,
given by the quantization {Wh := Tx(A)} of the center 3eit, is isomorphic to the
chiral algebra 930.
1.2
1.2.1
D-modules over the affine Grassmannian
The Beilinson-Bernestein Localization Theorem
Recall the theorem of A. Beilinson and J. Bernstein, that realizes 'D-modules on
the flag variety G/B as modules over the associative algebra given as the quotient
of U(g) by the maximal ideal of the center defined by its action on the trivial g-
17
module.
The main body of this second project tries to develop an analogue of
the above theorem in the affine case. More precisely, denote by GrG,z the affine
Grassmannian
GrG,x := G((t))/G[[t]
This is an ind-scheme of finite type classifying G-bundles on X with a given trivialization on X - x.
On the affine Grassmannian we can define the category
D-mod(Gra,x) of D-modules, and, as in the finite dimensional case, we are interested in describing it in different terms.
In particular, we would like to have
an algebraic description of it, where, by algebraic, we mean a description of it as
modules over some associative algebra. However, it turns out that the category that
can be realized as such is a critically-twisted version of D-mod(GrG,2). The reason
being that this new category is related to the category
eit-mod
and therefore to the
topological algebra Fun(Op4(D*)). This category, denoted by De,.it-mod(GrG,x) is
constructed in the following way. As it is explained in [BD2], there exist a canonical
line bundle
crit,x -+
GrG,x
on Gra,2. Critically twisted D-modules on GrG,x are just 0-modules on GrG,x with
an action of a particular sheaf 'Dc,
attached to Leit,2. The functor M -+ M 9
Lrit,x defines an equivalence of categories
D-mod(GrG,x) ~> Deit-mod(GrG,x),
(1.1)
therefore describing the RHS as modules over some associative algebra, would also
describe the category of D-modules on Grc,x as such.
We can start by consid-
ering the functor F of global sections on GrG,2 as a functor from the category
De,it-mod(GrG,x) to the category of vector spaces Vect. It can be shown that the
action of 'D
on a module M, gives a 'crit-module structure on the vector space
18
F(GrG,x, M). We therefore have a functor
I : Deit-mod(GrG,x)
crit-mod.
-+
However, unlike the finite dimensional case, this functor does not establish an equivalence of categories. It is not hard to see that F factors through the sub-category
Oc-it-modreg. However, the resulting functor F : Dc,it-mod(GrG,2)
-+
cit-modreg
is
not an equivalence. Instead, the following conjecture was proposed in [FG3]:
" The action of the groupoid IsomOp,(DX)
=
Op (D.) x Opb(D2) on 3,it lifts to
-|G
an action on scrit-modreg compatible with the action of G((t)).
" The functor F establishes an equivalence between Dcit-mod(GrG,2) and the
category
(Scrit-modreg)somo"(D-)of
IsomOp,(D.)-equivariant objects in 'crit-mdreg-
The above conjecture shows that F, viewed as a forgetful functor, does not realize
Derit-mod(GrG,x) as 3-mod, for some associative algebra 3.
In understanding how to describe the category De,it-mod(GrG,2), the questions that
arise are the following.
" Is there a different way of describing D-modules on GrG,x?
"
What do we mean by algebraic description?
An answer to the above questions is given by the notion of modules over a chiral
algebra. In fact critically-twisted D-modules on the affine Grassmannian can be
described as chiral modules for the chiral algebra D,it satisfying certain properties.
In the second project, we will define a chiral algebra 3 and a functor
: D,it-mod(GrG,x) -+ 3-modz,
19
which will realize the LHS as modules supported at x E X for the chiral algebra
B. We believe that the functor F],, defines an equivalence between the RHS and
a certain subcategory of '-modx
defined using the action of
3cit on Scit-mod.
However, we were only able to show the promised equivalence assuming a conjecture
concerning the functor Tx.
1.2.2
Construction of the functor
Consider the chiral algebra Ait. For this chiral algebra we have an equivalence
between the category Ac,-it-modx of Acrit-modules supported at x, and the category
,-it-mod introduced earlier. Let "ix denote the functor of semi-infinite cohomology,
which from now on will be simply called the Quantum Drinfeld-Sokolov reduction.
Tx : { Chiral Ac,.it-modules}
-+
{Chiral 3c,.it-modules}.
By the theorem of Feigen and Frenkel we have an equivalence
3c.it-mod, c- QCoh(Opb (D )) :=
{discrete continuous Fun(Opb (D*))-modules}.
If we restrict the functor 'Ix to the category A,it-modx of Ac,-it-modules supported
at x, we therefore have a functor
IF : Ait-modx -+ QCoh!(Op§(D*)).
Consider now the chiral algebra Dit of chiral differential operators on the loop
group G((t)). Recall the two embeddings
Acrit -+ Dcrit *- Acrit,
20
corresponding to left and right invariant vector fields on G((t)). Chiral Deit-modules
supported at x should be thought as D-modules on G((t)). In particular, if we denote
by 7r the projection
r: G((t))
-* GrG,x,
given a module M E Deit-mod(Gra,2), we can regard F(G((t)), wr*(M)) as an object
in Dit-mod2. We define Fp,2 to be
9crit-mod
F,, : D,it-mod(GrG,x,)
Denote by 3 the chiral algebra
(id N Px ('De,.it ).
93:
By construction, the action of
it on rp,2(M) can be lifted to an action of 3.
Recall now the ind sub-scheme Op "r of Op6(D'), and denote by QCoh' (Op") the
category of continuous discrete Fun(Opg";)-modules. We have the following.
Conjecture 1.2.1. The
functor Tpx establishes an equivalence of categories
D-mod(Gra,x)
~1_ Derit-mod(GrG,x,) ~_ 3-modunr,x
rq"
* Zcrit,z
where 2-modunr denotes the category of 3-modules supported at x E X, which are
supported on Op," when regarded as objects in QCoh (OpO(D )).
1.2.3
The factorization picture
In trying to prove conjecture 1.2.1 we immediately realized that we needed an understanding the categories involved as the point x moves. More generally, for n distinct
points X1 , ... ,xn on X, we need to understand the categories De,.it-mod(GrG,xi) 0
21
...
0 De,.it-mod(GrG,x,) and 3-modunr,x1 @
0
3
modunr,x., as the n points move,
hence, in particular, as they collide.
In the second project we will state a conjectural equivalence of categories over
any power X" of the curve. In particular it will imply the equivalence of conjecture
1.2.1, by "taking the fiber at x E X".
We will then explain how the conjecture
1.2.1 would follow from a conjecture concerning the functor XIx in its factorization
version, as explained below. The formulation of this conjecture uses the description
of chiral algebras in term of factorization algebras together with the notion of factorization spaces.
An important example of factorization space is given by the Beilinson-Drinfeld
GrassmannianGrG on Ran(X). This is given by the assignment
I -+ GrG,
where I is a finite set, and GrGJ is the space over X' given in the following way.
For an affine scheme S, an S point of GrGI consists of a map S
4
PG on Xs := S x X and a trivialization of PG onXs -
where 0, denotes
UiEIrFi,
XI, a G-bundle
the graph of the i-th component of 4 in S x X. In particular, for I = {1, ...
fiber of GrG,Xi at any
(i
1
, ..
., Xn ), with xi
, n},
the
# x, is the product of the corresponding
affine Grassmannians over each xi. This property of the Beilinson-Drinfeld Grassmannian is, indeed, one of the data in the definition of a factorization space.
On each GrGI there is a well defined notion of D-modules on it, and a well defined notion of critically-twisted D-modules. We denoted this category by De,.it-mod(GrG,I)The latter is given using a line bundle
I
cit
on GrG, i.e. a collection
-+ £Crit,1
22
of line bundles over GrGI. We will be interested in the factorization category
Dcrit-mod(GrG) given by the assignment
I
-4
Dcrt-mod(GrGI)-
As it is explained in [NR], given a chiral algebra A, we can define the category
A-mod, of chiral A-modules over XI.
The assignment I -+ A-mod, defines a
factorization category, simply denoted by A-mod. When A is commutative, in 3.1.1,
we will explain how to describe the category A-mod, as the category of modules
over a space
MAA
over XI,
M19A
-4 XI
canonically attached to A.
Consider the commutative chiral algebra A = 3,.it. We denote by Op' the
M 93 ct,} that should be thought as the
factorization space Op' = {I -+ Op,:
factorization version of opers on the punctured disc. For each I, the algebra of
function on Op", has a structure of topological algebra over X' and we have an
equivalence
QCohl (Op§, 1 ) := {discrete
continuous Fun(Opo,1 )-modules
I
~ 3,it-modr.
We denote by QCoh(Opo) the factorization category given by the assignment
I -+ QCohl(Opo,j).
For every I, we can define a certain sub-functor Opgn C
O pg,I
corresponding to the
space of unramified opers from the previous section. It can be shown that this subfunctor is represented by an affine-ind-scheme. This gives rise to a topology on the
algebra of functions Fun(OpuT ). As before, we will denote by QCoh(Op,)
23
the
category of discrete continuous Fun(Opg"/)-modules. By adapting the construction
of 1Ix to the factorization picture, we define functors
I : Acrit-mod, -+ QCoh1(OpO,I),
and use them to construct functors
F],
1
: Dcit-mod(GrG,I) -+ 3-mod(QCoh (Op',I)).
Using the above functors, we arrive at the formulation of the Ran(X)-version of conjecture 1.2.1.
Conjecture 1.2.2. The collection {I -+ 1ij, 1} together with the equivalence 1.1 give
rise to an equivalence of factorization categories
D-mod(GrG)
24
3-mod(QCoh(Opb"')),
rgo(.Lerit)
(1.2)
where 3-mod( QCoh ( Opu")) denotes the factorizationcategory {I -+ 3-mod(QCoh!(Op"r))}
of 3-modules on X'
which are supported on Opunj" when regarded as modules over
OpoI.
1.2.4
The main conjecture
As we mentioned before, the above conjecture, formally follows from a conjecture
concerning the functors TI. More precisely, consider the group Dx-scheme Jx(G)
as defined in 3.1.7. It acts on the category Acrit-modx of chiral Acit-modules on
X.
We can therefore consider the sub-category Acrit-mod
x(G)
of strongly JX(G)-
equivariant objects in Ac,.it-modx. For instance, if we consider Acit-modules supported at x, then the category Acrit-modfx(G) is the category
ing of
crit-modules
9 eit-modGtll
consist-
on which the action of g[[t]] can be integrated to an action of
G[[t]]. Consider the functor Tx,
Tx : Acit-modx -+ QCoh!(Opo,x).
24
This functor has being studied by D. Gaitsgory and E. Frenkel. In particular, in
[FG2] they show that, when restricted to the sub-category Ac,it-mod 'x(G) it defines
an equivalence
Acit-modx(G)
'Tx
QCoh (Op"').
(1.3)
Similarly to the above, as it is explained in 3.1.7 and 3.3.3, there exists a factorization
group JG = {I
-+
JG 1 } acting on the factorization category Acit-mod. We can
consider the sub-category A,.it-modiG of Acit-modules on XI, which are strongly
JG-equivariant, and consider the restriction of I1 to this category,
I : Acit-modiG -+ QCoh(Op"i).
The main conjecture is the following.
Conjecture 1.2.3. The collection I := {I
-+ IFI}
defines an equivalence of factor-
ization categories
Acit-modJG g
1.2.5
QCoh(Opnr).
How conjecture 1.2.3 implies conjecture 1.2.2
We will briefly explain how the equivalence in 1.2.3 would imply the equivalence
D-mod(GrG)
'3-mod(QCoh(Opnr)).
-+
r,o(o(Lrita)
As it is explained in 3.1.3, given a factorization category C we can define (chiral)algebra objects in C. Moreover, as it is explained in 3.1.8, for an algebra object A
in a factorization category C, we can define a factorization category A-mod(C) of
A-modules in C. Consider now the factorization algebra
Acrit-mod
JG
=
-
25
Acit-modiG}.
Using the right embedding Acrit -4 'Dit,
we consider 'Dit as an algebra object in
Acrit-mod JG. We can therefore consider the factorization category 'Derit-mod(Acrit-mod JG)
Similarly, we can consider the chiral algebra '3 = qx('Deit) as an algebra object in
the category QCoh!(Op "'). Conjecture 1.2.3 implies that we have an equivalence
of factorization categories
Deit-mod(Acit-mod JG) ~ 'B-mod(QCoh! (Opnr)).
(1.4)
In proposition 3.3.3 and in theorem 3.1 we will shown the following two facts.
" We have an equivalence 'Dit-mod(Acrit-modJG)
"
Tcrit-mod
V
JG
There exist an equivalence of factorization categories
Dcit-mod(GrG)
ODrit-modJG
Therefore the equivalence (1.4) can be written as
Dit-mod(GrG) ~+'B-mod(QCoh(Opg"r)),
which immediately implies the equivalence stated in 1.2.2 after tensoring with the
factorization line bundle
1.3
,it as explained before.
Organization of the thesis
e In Chapter 2 we start by recalling the definition of the basic objects that will
be used in this thesis. In particular, we will recall the classical definition of
chiral algebras. We will focus on commutative chiral algebras and on Lie*algebroids acting on them. For a commutative chiral algebra 'Z we will be
26
interested in studying Poisson structures on it. In particular, given a Poisson
structure on 'Z, in 2.2 we will define the notion of quantizations modulo h 2 of
it. This will be used to state the equivalence between such quantizations of
'R and chiral extensions of the Lie*-algebroid Q'('R) presented in [BD]. More
precisely, If we denote by Qch('R) the groupoid of C[h]/h 2-deformations of a
chiral-Poisson algebra 'R, and by '(Q'(R))
the groupoid of chiral extensions
of Q ('R), there is a functor
'Pch(Ql(Z))
(1.5)
+ Qch(R).
In [BD] 3.9.10. they show that the above functor is an equivalence. In section
2.3 we will consider the chiral algebra Acit and its center
3
cit.
We will define
a Poisson structure on it, and a quantization {Wr} of this Poisson structure.
The quantization will be constructed using the Drinfeld-Sokolov reduction Ix,
that will be introduced at the end of 2.3.1, as a special case of the BRST reduction, that will be studied in 2.3.1. We will then consider the chiral extension
Qc(3crit) of Q1(
3
cit) given by the above equivalence.
The main theorem of
this chapter will describe Qc(3eit) in terms of the chiral algebra 'Deit of critically twisted differential operators on the loop group, whose definition will
be recalled in 2.4.1.
More precisely, we will show that the chiral envelope
of Gc(3eit) coincides with the chiral algebra CO..t = ('Ix 0 xWx)('Do.it). The
proof of this theorem will rely on the explicit construction of the inverse to
the functor in (1.5) that will be given in 2.2.2. In fact, since the proof of the
equivalence 1.5 presented in [BD] does not provide such inverse, a large part
of this chapter will be taken by this construction. The last section will be devoted to the proof of theorem 2.3. In section 2.4.2 we will give an alternative
formulation of the Theorem that consists in finding a map F from Qc(3,it)
to C.,
with some particular properties. In section 2.4.3 we will finally define
27
the map F and conclude the proof of the Theorem.
* Chapter 3 is divided into 7 sections. The main conjecture 1.2.3 will only appear
in section 3.6, the reason being that its formulation needs some foundational
preliminary notions that will be given in the previous sections.
In section 3.1 we introduce the right categorical setting in which we will be
working. In particular, since conjecture 1.2.2 states an equivalence between
two factorization categories, we will define the notion of abelian category over
Ran(X) and the notion of factorization category. We will then define the notion of factorizationalgebra in a factorization category C and relate this notion
to the notion of chiral algebra. In 3.1.5 we will relate the notion of commutative chiral algebras to the notion of factorizationspaces. We will then address
our attention to the factorization category of modules over a chiral algebra A.
In 3.1.8 we will see how this notion plays out in the case of a commutative
chiral algebra.
In 3.2 we recall the definition of action of a group G on a category, and, in 3.3
we define the notion of action of a Dx-group-scheme 9 on a factorization category. In particular, in 3.3.1, we will study the action of the group Dx-scheme
9 of on the factorization category A-mod of chiral A-modules, as defined in
3.2. We will be interested in the category A-mod 9 of srtongly
S-equivariant
objects in A-mod. In 3.3.3, we will apply this to the group Dx-scheme Jx(G)
of jets into G, defined in 3.1.7, acting on the factorization category Dit-mod
of De,.it-modules.
In section 3.4 we recall the definition of the Beilinson-Drinfeld Grassmannian
GrG. This will be defined as a factorization space, i.e. we will have a space
GrGI over X' for every finite set I. We will then explain, in 3.23, how to
28
define the category D-mod(GrG,I) of D-modules on each GrGI. In 3.24, we
will then construct a line bundle Lcit,I over GrG,I and define the category
De,.it-mod(GrG,I) of critically-twisted-D-modules on GrG,I. In 3.4.4 we will
show how we can describe the category Dc,it-mod(GrGi) in terms of the category 'D.it-mod, of 'De,.it-modules on XI. In fact, we will show that the former
category is equivalent to the category of strongly Jx(G)-equivariant objects
in 'D,it-mod1 . We will then move to the definition of the factorization space
Op, corresponding to the chiral algebra 3,it. More precisely, in 3.5 we will
recall the definition of opers on the punctured disc Op,(D ) as given in [BD2]
and construct the factorization space Op' corresponding to it. We will then
define the co-unital factorization space Op. of regular oper, and the factorization space Op"', corresponding to opers on the disc, and to unramified opers
as defined in [FG2].
In 3.6 we will finally state the main conjecture from which we will derive conjecture 1.2.2. We will explain in details how to construct the factorization
functor {I
J1}, where
-+
IF,: Acrit-modr
-+
3c-it-mod, = QCoh!(Op,i),
denotes the Drinfeld-Sokolov reduction for modules over X' that will be explained in 3.6.1.
We will finally recall the equivalence (1.3) and state the
conjecture 1.2.3.
The last section combines together all the results from the previous ones to
finally come to the proof of conjecture 1.2.2. We will in fact use the DrinfeldSokolov reduction on X to define the chiral algebra '3 = Tx(Dit), then we
will use results from section 3.6 and 3.4 to first define a functor from Deit-mod
to the category '3-mod(QCoh!(Op')) of '3-modules in QCoh!(Op'), as defined
in 3.1.8. The equivalence showed in 3.1 between strongly-equivariant Jx(G)-
29
objects in 'Deit-mod and the category D,it-mod(GrG,x) will yield the factorization functor {I -+ L',i}, where Ip,i is
e,,i : Dcrit-mod(GrG,I) -+ '3-mod(QCoh'(Op,I)).
Assuming conjecture 1.2.3, we will finally show that the above functors induce
equivalences of categories
Dcrit-mod(GrG,I) ~ '3-mod(QCohl(Op,"r)),
rv,I
and this will conclude the proof of conjecture 1.2.2, and therefore of conjecture
1.2.1.
* The Appendix is devoted to an explanation of how we think conjecture 1.2.3
can be proven. We will present two different approaches. The first, presented
in A.1, consists in constructing a functor
I : QCoh' (Op"") - Acrit-modJG
and show that 4) and I1 are mutually inverse equivalences of categories.
The second approach, presented in A.2, consists in deducing the equivalence
Acrit-mod
0
~> QCoh'(Op,") over X' from the equivalence over X given in
(1.3). More generally, given a factorization functor G :e
-4
D between two
abelian factorization categories inducing an equivalence Gx : x
> 'Dx, we
will explain what conditions on it would guarantee equivalences G, :
'Dx over XI.
30
4x
~
Chapter 2
W-algebras and chiral differential
operators at the critical level
2.1
Chiral algebras
We start this chapter by introducing the notion of chiral algebra as presented in
[BD].
We will see later, in section 3.1.3, how chiral algebras can be described as
factorization algebras, i.e. a sequence of quasi-coherent sheaves on X" satisfying
some properties. Since we will only use the latter description in the second chapter,
we prefer giving the classical definition here. Throughout this chapter A : X
X x X will denote the diagonal embedding and
j
:U
-4
"
X x X its complement,
where U = (X x X) - A(X).
For any two sheaves M and X denote by MMN the external tensor product 7r*M 0
OXxX
7r2N,
where 7ri and
7r2
are the two projections from X x X to X. For a right 'Dx-
module M define the extension A,(M) as
Al(M) := j*j*(Qx Z M)/Qx Z M.
31
Sections of A!(M) can be thought as distributions on X x X with support on the
diagonal and with values on M. If M and X are two right 'Dx-modules, we will
denote by M 0 N the right 'Dx-module M 0 N 9 Q*.
We will now recall the definition of unital chiral algebras as presented in [BD].
Definition 2.1.1. A unital chiral algebra A is as a right 'Dx-module Ad on X
equipped with a Dx-module homomorphism
p : j~j *(Ac' Z Ac) -+ A, (Ac)
where j:
X x X - A(X) -+ X x X +- X : A, and an embedding
i : Qx "- Ac'
satisfying the following conditions:
" (skew-symmetry) y = -o12 0 P 0 0-12.
"
(Jacobi identity) pi{2 3 }
"
(unit) The following diagram commutes:
= P{ 1 2 13
+
/2{13}-
j*j*(92x MA')
I
A i(Ac')
>
i
j~j*(Ad X Ad)
0.Ai
I,
(Ac)
where the vertical map on the left comes from the sequence
£x ZAdA-
and
j~j*(Qx Z A)
-+ A!A'(Qx Z Ad)[1] ~ A,(Ad)
-12 is the induced action on A by permuting the variables of X 2.
32
The Jacobi identity above means the following: if we denote by
of the subset of X
3
j123
the inclusion
where all the x's are different and by Aij the inclusion of the
diagonal xi = xy, then pi{23} :
j~j *(Ad
Z Ad Z Ad)
-4
A, (Ad) is defined as the
composition
j~j*23 (Ad
Z Ad Z Ad) 4
xj*
(Ad Z A( 2 3 )!Ad)
A(23)!i**
(Ac Z Ac) 4
j
~
(Ad),
A(123 )!
the map p1{12}3 is the composition
j~j*23(A" Z Ad Z Ad) 4
j*,j*(A(1
2 )!Ad
~ A(12)! (j*jX2
3
(Ad
N
E Ad)
Ae)) 4
A(123)!(Ac),
and the map p2{i 3 } is gives as
jji23 (Ad
Ad
A)
4
* 2 X3 ,XX (A(13)!Ad
~
A(13)
*
Z Ac)
(Ac' Z Ac') 4
A( 1 2 3 )! (Ac).
The Jacobi identity means that, as a map taking place on X 3 , the alternating sum
of the above maps is zero.
2.1.1
Commutative chiral algebras
As in the world of classical algebras, there is a well defined notion of commutative
chiral algebra. We will see how these are the same as affine Dx-schemes. Moreover,
in 3.1.6, we will relate the factorization description of commutative chiral algebras
to the notion of co-unital factorization spaces.
Definition 2.1.2. Let (2, pu) be a unital chiral algebra. T is called commutative if
33
the composition
'R M '9
>
j~j*(' z 'R) 1
>
A,('R)
(2.1)
vanishes.
Denote by T' the left 'Dx-module given as 'ZR:= 'R 0 £* . The diagram in (2.1)
implies, and is in fact equivalent, that t factors through a map
j~j*('R z 'R)
>
A
A('R)
Therefore we obtain a map
0'
This map yields a commutative product (because of the skew-symmetry) 9'R9
m+
z', making JZ' a 'Dx-algebra. On the other hand, if we are given a 'Dx-algebra 'R',
i.e. a left 'Dx-module with a map of Dx-modules JZ'
'Z := (')'
R' -+ 9', we can consider
and the composition
j*j*(RZ Y') = jj*(g' Z ') 0 j*j*(Ox Z Qx) 4 AiAI(9RZ
= A,('
0 Z'I)0 A,(%x)
'R) 0 Ai(Qx)
=
AId ) 0 A,(Qx) =
where m is the product map of 9'. This is a chiral operation on 'R. The above
establishes an equivalence
{Dx-algebras Z'}
+
{Commutative chiral algebras 'R}.
(2.2)
For instance, in the case 'R = Ox, with chiral product defined as p(f(x, y)dxZdy) =
f(x,y)dx A dy (mod G'X), you simply recover the commutative product on the
sheaf of functions on X, which is in fact the left 'Dx-module corresponding to Ox.
34
2.1.2
Chiral envelope of Lie*-algebras
The chiral algebras that we are mostly interested in are those that can be constructed
from Lie* algebras by taking their chiral envelope. Let L be a Lie*-algebra, as
introduced in [BD]. A Lie*-algebra L is, in particular, a right Dx-modules with a
map
[,IL : L M L
-
A,(L),
satisfying certain properties. The natural embedding
M M
c
j~j*(M0M)
defines an obvious forgetful functor
{ chiral algebras }
-+
{Lie*-algebras}.
We will denote by ALie the Lie*-algebra corresponding to the chiral algebra A. The
above functor admits a left adjoint U,
U : {Lie*-algebras} -+ {chiral algebras }.
Given a Lie*-algebra L, we define its chiral envelope to be the chiral algebra U(L).
In particular, by definition, we have
Homch(A, U(L)) ~ HomLie (A Lie, L).
The chiral algebra U(L) is generated by the image of L which is a Lie*-subalgebra
of U(L). The corresponding filtration on U(L) is called the Poincare'-Birkhoff-Witt
filtration. We have a canonical surjection
Sym'L -* gr.U(L).
' 35
As it is explained in [BD], when L is Ox-flat, the above surjection is an isomorphism.
2.1.3
The chiral algebra A,
Let g be a simple finite dimensional Lie algebra. Recall the Lie*-algebra L. = go'Dx
as defined in [AG]. For every symmetric invariant bilinear form
g
g
-
C,
consider the pairing ;,
:(
(Ox)g
Ox)
-
(a, b)
and extend this pairing to a map Rx:
KDX
Qx
(da, b),
-+
L 9 0 Lg
-+
A,(Qx 9 'Dx). The composition
of Rex with the map Qx 0 'Dx -+ Qx defines a 2-cocycle on the Lie*-algebra
L.. We defineL' to be the Lie*-algebra extension corresponding to this cocycle.
We will denote by A, the twisted chiral envelope of L,
At := U(L")/1 - 1,
where 1 denotes the embedding of Qx given by the identity in U(L') and 1 denotes
the embedding of Qx given by the construction of L,.
Remark 2.1.1. Given a bilinear form n, we can consider the Lie-algebra extension
9,, given as
0
-
C - 1 -+ gs
-
g((t))
-
0,
with bracket given by
[af(t), bg(t)] = [a, b]f (t)g(t) + r(a, b)Res(f dg) - 1,
36
where a and b are elements in 9, and 1 is the central element.
Recall that we denoted by U' the appropriately completed twisted enveloping aland by i-mod the category consisting of i-modules M on which the
gebra of i,
central element 1 acts as the identity and such that, for every m E M, the action
of atN on it is zero for N >> 0. Consider the chiral algebra A,
and the category
AK-mod2 of A-modules supported at x E X. Recall the associative topological algebra A attached to A, with the property that its discrete continuous modules are
the same as A-modules supported at x. When we take A to be AK, the topological
associative algebra A,; is isomorphic to U.
In particular we have an equivalence
of categories
gi-mod ~ A-modx.
We will be interested in the critical level r. = rit:=
the chiral algebra A,,
and by
3eit
-1/2KKill.
Denote by Acrit
its center. The importance of choosing the
level r, to be rit relies on the fact that the center Z(4erit-mod) of the category
rit-mod
:= U',.it-mod, happens to be very big, unlike any other level r : r,it
where the center is in fact just C, as shown in [FF]. The chiral algebra 3erit is
closely related to the center Z(eit-mod), in fact we have an isomorphism
3cit,2 ~ Z(eit-mod).,
in particular Z(-erit-mod)-mod is equivalent to the category 3cit-modx of
(2.3)
3cit-
modules supported at x.
2.1.4
Lie*-algebroids and 'R-extensions
Let 9Z a commutative chiral algebra. In this section we will recall the definitions of
Lie*-9Z algebroids and chiral J-extension of such. These definitions will be used in
2.1.6 to define the notion of chiral envelope of a chiral 9J-extension of an algebroid
37
Let ('Z, m : Z0 'R -+ 'R) be a commutative chiral algebra. Given a Lie* algebra L,
we say that L acts on 'R by derivations if we are given a Lie* L-action on 'Z
T : 'R 0 'R -+ A,('R),
which is a derivation of the product m.
Definition 2.1.3. Let L be a Lie* algebra acting by derivations on 'R via a map
T.
An 'R-extension of L is a 'Dx-module Lc fitting in the short exact sequence
0 - 'R +Lc > L -+ 0
together with a Lie* algebra structure on Lc such that -r is a morphism of Lie*
algebras and the adjoint action of Lc on 'R C Lc coincides with r o r.
Definition 2.1.4. A Lie* 'R-algebroid Z is a Lie* algebra with a central action of
'Z ( a map 'R 0 Z - L) and a Lie* action rL of L on 'T by derivations such that
e
T
is 'R-linear with respect to the £-variable.
* The adjoint action of Z is a Tr-action of Z (as a Lie* algebra) on Z (as an
'-module).
In the next definitions we consider objects equipped with a chiral action of 'z instead
of just a central one.
Definition 2.1.5. Let R be a commutative chiral algebra, and Z be a Lie* 'Ralgebroid. A chiral '-extension of Z is a Dx-module L' such that
0 - 9z 4 ZN
e
38
L -+ 0,
(2.4)
structure p,c
together with a Lie* bracket and a chiral '-module
on Z' satisfying
the following properties:
" The arrows in (2.4) are compatible with the Lie* algebra and chiral '-module
structures.
" The chiral operations pg and pgz,rc are compatible with the Lie* actions of L'.
" The * operation that corresponds to pars
the restriction of pz,,c to
(i.e.
'Z 0 Z') is equal to -i o o- o rzcz o -, where Tc,g is the Zc-action on 'Z given by
the projection Z'
-4 Z
and the Z action
T3,_
on 'Z and o is the transposition
of variables. In other words the following diagram commutes
C
'R
z
A,(i)C
>
j*(R9
0
t z)
,(e
Remark 2.1.2. The triples ('R, L, Lc) form a category in the obvious manner. For
fixed 'Z and Z, the chiral 'R-extensions of Z form a groupoid, denoted by
It is important to notice that 'Peh(Z) is not a Picard groupoid.
'(L)
The notion of
trivial chiral R-extension of L makes no sense. However, if we denote by 'Pc(Z)
the Picard groupoid of classical L-extensions, i.e.
extensions in the category of
Lie*-algebroids, then we have that the Bear difference of two chiral extensions is a
classical one, therefore we have the following.
Proposition 2.1.1. If 'Ph(L) is non-empty, then it is a 'c(L)-torsor.
Definition 2.1.5 can be extended by replacing 'R with any chiral algebra
dowed with a central action of 'R. More precisely a
39
e
en-
chiral C-extension of L is a
'Dx-module Z' such that
0 -+
(2.5)
C -+ Lc -+ L -+ 0,
together with a Lie* bracket and a chiral 'R-module structure
[tjc
on Lc such that:
" The arrows in (2.5) are compatible with the Lie* algebra and chiral '-module
structures.
" The chiral operations ye and ptZc are compatible with the Lie* actions of Lc.
" The structure morphism 'R -+
C is compatible with the Lie* actions of e.
" The * operation that corresponds to pryc (i.e.
[Lsc
restricted to 'R M Lc) is
equal to -ioo-oTjc,3oU, where rzcj is the Lc-action on 'R,
a
is the transposition
of variables and i is the composition of the structure morphism JZ -+ C and
the embedding C C Le.
Definition 2.1.6. The chiral envelope of the chiral extension ('R, C, ze, L) is a pair
(U(e, Lc),
4c),
where U(C, Lc) is a chiral algbera and
#'
is a homomorphism of Lc
into U(C, Zc), satisfying the following universal property. For every chiral algebra
A and any morphism
f
: Lc -+ A such that:
e
f
is a morphism of Lie* algebras.
e
f
restricts to a morphism of chiral algebras on C C Lc.
e
f
is a morphism of chiral-'-modules (where the '-action on A is the one given
by the above point),
40
there exist a unique map
f: U(C, LC)
-+ A that makes the following diagram
commutative
Le ---L _>-A .
U(e, Zc)
It is shown in [BDI that such object exists. When C =
we will simply write U(Lc)
instead of U('R, Le).
2.2
Quantization-deformation of commutative chiral algebras
Definition 2.2.1. Let 'R be a commutative chiral algebra.
R is called a chiral-
Poisson algebra if it is endowed with a Lie*-bracket, called the chiral-Poissonbracket
{, } :R
T'R
-
A,(')
that is a derivation of 'R in the sense of 2.1.4.
Example 2.2.1. Let At be a one-parameter flat family of chiral algebras; i.e., At is a
chiral k[t]-algebra which is flat as a k[t] -module.
Assume that A := At=o := At/tAt
is a commutative chiral algebra. This means that the Lie*-bracket [, ]t of At is
divisible by t. Thus
{, }
:= t- 1 [, ]t is a Lie*-bracket on At. Reducing this picture
modulo t, we see that A is a chiral-Poisson algebra, with bracket
{, } := {, }=0
One calls At the quantization of the coisson algebra (A,
parameter t.
41
{, })
with respect to the
2.2.1
Quantizations of chiral-Poisson algebras
As in the usual Poisson setting, one can consider quantizations mod tn+1 , n ;> 0,
of a given chiral-Poisson algebra (A, {, }). Namely, these are triples (A(), {, }(n)),
where A)
is a flat chiral k[t]/tn+1-algebra, {, }(n) a k[t]/tn+'-bilinear Lie*-bracket
on A(n) such that t{, } equals the Lie*-bracket for the chiral algebra structure, and
a : A(n)/tA(n)
-+
A an isomorphism of chiral algebras that sends {, }(n) (mod t) to
{, }. Quantizations modulo
tn+ 1
form a groupoid.
Now, let 'Z be a commutative chiral algebra. As it is explained in [BD] 1.4.18,
a Poisson structure on 'R gives the module Q ('R) a structure of a Lie* algebroid.
In fact, the bracket {, } yields a Lie* 'R algebroid structure on 'R
easily that the kernel of the projection 'R 0 9Z
'
0
-4
1 ('R),
'R. One checks
a 0 b -+ adb, is an ideal in
'R, therefore Q1 (') inherits the Lie*-'R algebroid structure.
Now consider the following: given a chiral extension
0 -+ 'R
-4 Oc(,)
-4Q1('R) -+ 0,
consider the pull-back of the above sequence via the differential d : 'R -+
Q1('R).
The resulting short exact sequence is a C[h]/h 2 -deformation of the chiral-Poisson
algebra
'.
If we denote by Qch(')
the groupoid of C[h]/h 2 -deformations of the
chiral-Poisson algebra 'R, and by Pch(01('R)) the groupoid of chiral 'R-extensions of
Ql
(') as defined in 2.1.2, the above map defines a functor
TchQ19Z)
_
Qch(Z).
In [BD] 3.9.10. the following is shown.
Theorem 2.1. The above functor defines an equivalence between ych(Q1('R)) and
'If {, } denotes the Poisson bracket on 'R, this is indeed a quantization of (', 2{, })
42
Qeh(R).
The above equivalence is the point of departure of this work.
Quantization of the center 3,it
Recall the commutative chiral algebra
3
,it defined in 2.1.3. We will see later, that
the Drinfeld-Sokolov reduction qIx, introduced in 2.3.1, produces a 1-parameter
family of chiral algebras {Wh} :{=
Tx(Ah)} .such that WO ~
2.2.1, we therefore have a Poisson structure on
{
[ih,
}
1=
where z = zala--o and w = ivala=o.
wh'wh
h
3
,.it.
According to
3,it defined by
(mod h)
It follows from the definition of the functor
9x that this Poisson structure coincides with the one from 1.1. Consider now the
following diagram:
Pch(Q1('Z))
Qch(jZ)
: {Lie* algebroid structures on
1(2)
: {Chiral-Poisson structures on 'R}.
A natural question to ask is the following: if we consider the quantization of
3,it introduced before, how does the corresponding chiral extension of Ql( 3
,.it)
look like?
The answer to the above question is the main body of this chapter.
2.2.2
Construction of the inverse
In this section, we will give an explicit construction of Qc('R) for an arbitrary chiralPoisson algebra JZ. In the case where 'Z =
43
3,it we will see how this chiral extension
relates to the chiral algebra of differential operators on the loop group G((t)) at the
critical level introduced in [AG], where G is the algebraic group of adjoint type
corresponding to g.
Let R be a commutative-Poisson chiral algebra and {'Rh} a quantization of the
Poisson structure. The equivalence of categories from Theorem 2.1 states the existence of a chiral extension
0 -+ 2 -+ Q'('R) + Gl('R) -+ 0
However the proof of this theorem doesn't provide a construction of it. This section
will be devoted to the construction of the above extension.
Starting from the Lie* algebra extension
0 -+ 'R -+ ' -+ 'R -+ 0,
where
Z' :='JZ/h 2 'Ra
acts on 'R via the projection R' -+ R and the Poisson bracket
on 'R, we will first construct a chiral extension (see Definition 2.1.5) Indh(Jc) fitting
into
0
' -+ Indc('Rc) -+ 'R 0 R - 0,
where 'R 0 'R is viewed as a Lie* algebroid using the Poisson structure on 'R. The
chiral extension QG('R) will be then defined as a quotient IndRJZc).
More generally, in 2.2.2-2.2.2 we will explain how to construct a chiral extension
Ind (Lc) fitting into
0 -'Z -+ Ind (Lc)
-4
'R &L -+ 0
for every Lie* algebra L acting on 'R by derivations and every extension
44
(2.6)
The case where L = 'Z and Lc
9Zc, will be presented in 2.2.2 as a particular case
=
of the above general construction.
Definition of Indh(L)
Let ('R, p) be a commutative chiral algebra and let L be a Lie* algebra acting on 'R
by derivations via the map r. The induced 'R-module 'R 9 L has a unique structure
of Lie* '-algebroid
such that the morphism 1gz
0 idL : L -+ 'R 0 L is a morphism
of Lie* algebras compatible with their actions on 'R. Note that we have an obvious
map
i : L -+ ' & L.
The Lie* algebroid 9Z 0 L is called rigidified. More generally we have the following
definition.
Definition 2.2.2. A Lie* algebroid Z is called rigidified if we are given a Lie* algebra
L acting on 'R via the map T, and an inclusion i : L -+ Z, such that 'R 0 L + Z.
Let Z be a rigidified Lie* algebroid. Consider the map that sends a chiral extension of Z
.0
4'R
9z-
L' - L -+ 0
to the 'R extension of L given by considering the pull-back of the map i : L -+ L.
Denote by 'Pc(Z) (resp. 'Ph(Z)) the groupoid of classical (resp. chiral) extensions of
L (where by classical we mean extensions in the category of Lie* algebroids), and by
P(L, r) the Picard groupoid of 'R-extensions of L. Clearly the map mentioned above
(that can be equally defined for classical extensions as well), defines two functors
'd(L)
-+ 'P(L, r),
'chP()
-+ 'P(LT
As it is explained in [BD] 3.9.9. the following is true.
45
Proposition 2.2.1. If L is Ox flat, then these maps define an equivalence of
groupoids
d(L)
~4 'P(LT)
T (LT)
ych(L)
Given a Lie* algebra extension 0 -+ 'R -
Lc
-+
(2.7)
L -+ 0, define Ind" (Lc) (resp.
Indc(Lc)) to be the classical (resp. chiral) extension corresponding to the above
sequence under the equivalences stated in the above proposition.
In 2.2.2 we will briefly recall the construction of the inverse functors to (2.7) in
the classical and chiral setting respectively (as presented in [BD]). However in 2.2.2
we will give a different construction of the inverse functor in the chiral setting, i.e. a
different construction of the chiral extension Ind"' (LC) associated to any 'Z-extension
of L. The latter construction will be used to define the chiral extension Qc(,R)
The classical setting
For the "classical" map 'M(L)
-
'P(L,
T),
0 4 '-+
to an extension
Lc -+ L -+ 0,
(2.8)
the inverse functor associates the classical extension Ind (Lc) of the Lie* algebroid
'Z 0 L = L given by the push-out of the extension
0 -9Z
'R
'R
2
J'o
Lc -+ 'R (&L - 0
via the map m : 'Z D JZ -+ 'R.
The construction of the inverse functor in the "chiral" setting given in [BD] (i.e.
the construction of Indc(Lc)), uses the following two facts:
46
.
T'Pc()
has a structure of 'PT"(Z)-torsor under Baer sum.
*
'hP(Z)
is non empty.
The first fact follows from condition 3) in the definition of chiral :-extension, which
guarantees that the Baer difference of two chiral extensions is a classical one. In
other words the action of 9Z on the sum of two chiral extensions is automatically
central.
The non emptiness of 'ch
(L) follows from the existence of a distinguished chiral
Such object is
9Z-extension Indgz(L) attached to every Lie* algebra L acting on 9.
defined by the following:
Definition-Proposition 2.2.1. Suppose that we are given a Lie* algebra L acting
by derivations on 3Z via the map
T,
and let Z be a rigidified Lie* algebroid (see
Definition 2.2.2), so we have a morphism of Lie* algebras i : L
9Z 0 L
~+ Z.
-4
L such that
Then there exist a chiral extension Indjz(L) equipped with a lifting
i : L -+ Indjz(L) such that i is a morphism of Lie* algebras and the adjoint action
of L on R via i equals r. The pair (Indiz(L),7) is unique.
The proof of this proposition can be found in [BD] 3.9.8. However in 2.2.2 we
will recall the construction of Inda(L) and of the map i: L -
Indjz(L).
To finish the construction of Indh(Lc) (or in other words, the construction of the
inverse to the functor 'Ph(9)
-
'(L, r)), we use the classical extension Ind" (LC)
given in 2.2.2 together with the '(L)-action
on Ph(Z). To the extension 0
Lc -+ L -+ 0 we associate the chiral T-extension
Indh(Lc) := Ind (Lc) + Inda(L)
Baer
47
-+
9-
of 'JZ 9 L by 'Z, where IndT(L) is the distinguished classical extension defined in
2.2.1. Note that, after pulling back the extension
Ind
0-+ '
via the map L
-+
Z ~ 9'R
h(Lc) - Z ~ 'R 9 L -+ 0
L, we obtain the Baer sum of the trivial exten-
sion (corresponding to Indg(L)) with LC, i.e. we recover the initial Lie* extension
0 -+ 'Z
-
Lc -+ L
-+
0 as we should.
Construction of Indj(L).
In this subsection we want to recall the construction and the main properties of the
distinguished chiral extension Ind)Z(L) given by Definition-Proposition 2.2.1.
Given a Lie* algebra L acting on 'R by derivations, we can consider the action
map 'Z M L
->
0
0
Ai('R) and consider the following push out:
> j~j*('R Z
'RM L
>
If
Ai('T)
>,
L)
4
> A('
0 L)
0
1
('R) @j,j*('Z Z L)/'R Z L ->- At ('R 9 L)
>0.
The term in the middle is a 'Dx-module supported on the diagonal, hence by Kashiwara's Theorem (see [?] Theorem 4.30) it corresponds to a 'Dx-module on X. This
'Dx-module has a structure of chiral extension and will be our desired Indg(L) (i.e.
we have Ai(Indz(L)) : A,(R) Djj*('Z 0 L)/'R Z L).
Remark 2.2.1. By construction we have inclusions 'Z
48
-+
Indz(L) and a lifting
i: L
0
-+
>
Indg(L) of i
x
L -+ 'R 0 L. In fact we can consider the following diagram
:1j~j*(Qx Z L)
0 L
L) ~_A, (L)
>A(Qx
I
: j~j*('Z Z L)
A ('R
:A('2) D j~j*(', Z L) /'R ZL
A '
> 'R
0M L
0
i(R
>0
Ai(i)=At(unitgid)
L)
0
)>0.
By looking at the composition of the two vertical arrows in the middle, it is not hard
to see that this composition factors through A, (L). In fact the most left vertical
arrow from Qx Z L to A ('Z) is zero. We define i to be the map corresponding (under
the Kashiwara's equivalence) to Ai (i).
As it is shown in [BD] 3.3.6. the inclusions 'R -+ Indg(L), i: L
chiral operation
j~j*('
Z L)
-+
-+
IndT(L) and the
Ai(IndJz(L)), uniquely determine a chiral action of
'R on Indjz(L) and a Lie* bracket on it. In other words they give Ind-T(L) a structure
of chiral 'R-extension.
Note that this chiral 'R-extension corresponds, under the equivalence given by Theorem 2.2.1 (i.e. after we pull-back the extension via the map I/x : L -+ '9Z 0 Z), to
the trivial extension of L by 'R in '(L,
T).
To summarize we have seen that:
" If a Lie* algebra L acts on 'R we can construct the distinguished chiral extension IndjT(L) of Z with a lifting i : L
-+
IndT(L) of the canonical map
i : L -+ Z.
" From an extension 0 -+ 'Z -+ L' -+ L
-+
0 we can construct a chiral extension
Indh(L') with a map L' -+ Ind h(Lc) given by the pull-back of L -+ L.
49
Remark 2.2.2. Clearly, if we have the extension 0
-+
'Z -+ Lc -+ L -+0, we
can also consider Lc as a Lie* algebra acting on 'R via the projection Lc
-+
'R. In
other words we forget about the extension and we only remember the Lie* algebra
Lc.
From point one of the above summary we can construct the distinguished
chiral extension Ind(Lc) corresponding to this Lc action on 'R, together with a map
i : Lc -+ Indz(Lc).
Different construction of Indch(Lc).
We will now explain a different construction of the chiral extension
0
'
- Ind (Lc)
- L ~' 9
L -+ 0
that will be used later to construct Qc(JZ).
As it is explained in the Remark 2.2.2, given an 'R-extension
0 -+
Lc -+ L -+ 0,
we can consider the action of Lc on 'Z given by the projection Lc
-+
L and construct
the distinguished chiral extension Indg(Lc). This is a chiral J-extension fitting into
0
-+
A('
-+
A,(Indj(Lc)) -+ A,('R
0 LC) -+ 0,
where A,(Indz(Lc)) ~ A(R) ejj*('Z X LC) /'R M Lc. Since we ultimately want an
extension of 'R by 'R 0 L, we have to quotient the above sequence by some additional
relations. We will in fact obtain Ind h(Lc) by taking the quotient of Indg(Lc) by the
image of the difference of two maps from R 0 T'
50
-+
Indgz(Lc).
The above maps are given (under the Kashiwars's equivalence) by the following two
maps from Ai('R 0 9') to Ai (Ind,(Lc)).
1) The first map is given by the composition
A ('R (&'R) -M+A 1('R)
A 1(IndT (Le))
2) For the second map, consider the following commutative diagram:
0
>Z'R
'R
9Z
>
I
jidOk
0
0
>
I
A,('R)
>
>l A('R
klid
:j~j*('R 0 L)
'RZ Lc
>
j*('R Z ')
1r
A('Z) Dj,j*( M Lc)/'Z M LC
A(Indj (Lc))
@'R9)
>0
jidek
A (' 0 Lc)
0
A ('R 0 Lc)
0.
4
We claim that the composition of the two vertical arrows in the middle (i.e. 7r o
(k Z id)) factors through A,('R 0 9Z). In fact since the action of L' on 'R is given
by the projection Lc -+ 'R, the copy of 'Z inside L' via k acts by zero. Hence the
composition of the left most vertical arrows is zero, which shows that there is a well
defined map
k :A
0(R ) -+ A,(Inda (L')).
The quotient of Ind_(Lc) by the image of the difference of the above maps is exactly
Ind (Lc).
Remark 2.2.3. Note that the inclusion Lc -+ Ind 1(Lc) mentioned in the summary
in 2.2.2 corresponds to the composition
A,(Lc)
=)
AI(Ind,(Lc)) -w Ai(Ind'(Lc)).
51
(2.9)
A special case: deformations of 'R.
Let ('R, m : R 0
R -+ 'Z) be a commutative chiral algebra given as 'R := 'R,/h'R,
where {'R,1} is a family of chiral algebras. Denote by
{, }
the Poisson bracket on '
defined as
{z,w} =
[zh,wr]n
(mod h),
where zh = z (mod h), wr, = w (mod h) and [,
Jr
denotes the Lie* bracket on 'Ra
induced by the chiral product pr restricted to 'R, M 'R.
Consider the quotient
J' = 'Z /h'9r.
This is a Lie* algebra with bracket
[,
]c
defined by
Consider the short exact sequence
0
'T
T'--+ 0,
and let us regard 'Rc as a Lie* algebra acting on '
(2.10)
via the projection 'ZC -+ '
followed by the Poisson bracket multiplied by 2 1/2. This sequence is an '-extension
of 'R in the sense we introduced in Definition 2.1.3, therefore, from what we have
seen in 2.2.2, we can construct a chiral '-extension of 'Z
'R by 'R (here L = 'R and
LC ='ZC)
0~
c pIdi'c)
-+' -+ 'R @'R -Z+0.
(2.11)
Below we will use the above chiral extension to define the chiral algebroid Qc(9g).
The construction of
c('R).
We can now proceed to the construction of Qc('). Recall that, because of the Poisson bracket on 'R, the sheaf Q1 ('R) acquires a structure of a Lie* algebroid.
2
This correction is due to the fact that, as we saw in 2.2, the equivalence stated in Theorem
2.1 gives a quantization of 1/2{, }.
52
Recall that we denoted by Qch('R) the groupoid of C[h]/h 2-deformations of our
chiral-Poisson algebra 'R, and that we want to understand how to construct the
inverse to the functor
'Pch
pi(3Zg _4 genga)'
that assigns to a chiral extension 0 -+R -+
the differential d : 'R -
c(JZ) _+ gi('R) -+ 0, its pull-back via
Q1('R).
The inverse functor will be constructed as follows: for any object in QGh(JZ), i.e.
to for any extension 0 -+ 'R - 'Rc -+ 'R -+ 0, we will consider the chiral extension
0 -+ 'R -+ Ind ('Rc) -+ 'R
'R -_+0
described in the previous subsection. We will quotient Ind ('zc) by some additional
relations in order to impose the Leibniz rule on 'R 0 'R.
These relations will be
given, under Kashiwara's equivalence, as the image of a map from Ai(9c 0 ,Rc)
to A!(Ind ('Rc)).
More precisely, we will construct a map from j~j*('Zc Z 9')
to A!(Indg('Rc)) such that the composition with the projection A!(Indz('Rc))
A,(Indc(Rc)) vanishes when restricted to 'ZC N 'Rc.
A,(Zc 9 9Ze)
-+
Hence it will induce a map
Ai(Ind h('Rc)). Form the sequence (2.11) we will therefore obtain a
chiral 'R-extension Qc(9) of the Lie* algebroid Q'('R)
0 -4 'R' -+ Q ('R) -+ Q1('R) -4 0.
We will then check that 'Z',
which a priori is a quotient of 'R, is in fact 'Z itself, and
that the pull-back via the differential d: 'R -+ Ql('Z) is the original sequence (2.10),
with induced Poisson bracket given by
{,}. This will imply that Qc(R) is in fact the
chiral extension Q'('Z) given by Theorem 2.1.
53
The map from
j,j*('RCNJ)
to Ai (IndT(9U)) is defined as the sum of the following
three maps:
1. The first map a1 is given by the composition
j~j*( c N 'Rc) -4 ja* ('R z 'Rc) -+ A, (Indoz(9Zc)
where the first map comes from the projection 9ic
R
-4
2. The second map a2 is obtained from the first one by interchanging the roles
of the factors in j~j*('Rc Z 'c).
3. For the third map a3, note that the chiral bracket pa on 'Zh gives rise to a
map
-pc : j~j*(g
Z 'Rc) -4 A, ('RC)
and we compose it with the canonical map A!(Rc) -+ A(Ind,('Rc)).
Now consider the linear combination ai - a2 -
03
as a map from j~j* (Tc Z ,c) to
Ai(Indg('Rc)). If we compose this map with the inclusion 9Z Z
J'
-+ jj*('c
9,c)
and the projection onto Ind h ('R,), it is easy to see that the map vanishes. More
precisely we have the following:
Lemma 2.2.1. The composition
9Z Zc
~" jKj*(JzC 0
j~c)
al-E?2-013) A!(IndX~c))
vanishes. Thus it defines a map Leib: A,(9Zc 0
eZe)
-
!(n~~R)
_A (Ing f(c)).
Proof. Since the action of 9' on 'R is given by the projection 9Rc -+ 'T and the Poisson
bracket on R multiplied by 1/2, and because of the relation o- o {, } o o- = -{,
54
},
the maps ai and a2 factor as
'Ze
Z C
A (Ind('c))
>
A (Indh('c))
Note that the above wouldn't have been true if we hadn't used the relation in
Ind(Z') as well. Moreover the third map, when composed with the projection to
A,(Ind ('Ze)) is exactly
'RC 0 'R - 'R Z 'a !'-4 A, (') -* A,(Indc('c)),
hence the combination a 1
-
a2
-
a 3 is indeed zero. From the above we therefore get
0 9c) -+ Ai (Ind('Rc)).
a map A,('
We define Q2
) to be the quotient of Indh(Rc) by the image of the correspond-
ing map from Tc 0 9Re to Ind ('Rc) under the Kashiwara's equivalence.
'Rc) indeed factors through
Remark 2.2.4. Note that the map Tc 9 'c -+ Ind
'Rc
'0
0 9Ze -* 9Z
9. To show this it is enough to show that the map
factors through j(j*('Rc 0 9c) -
A! (Indc ('))
j,j*('Z 0
j j*Z(C
q'RC) 4
'R). If so, then the diagram
below would imply that the composition 'Z Z R -+ Ai (Ind (c)) is zero, and we are
done:
0
> 'RZ
9Zc
,
g
,(,gc
N 9Zc)
Ai
Idc
, )
1'Id4
)
55
,
,(9,e
(9
-ge)
>0
.
To show that the map factors as
j~j*(9'R ')
A (Indh(Rc))
j,j*(R M 'R)
we need to show that the composition of the map ai - a2 - a3 with the two embeddings j~j*(9' M 'R) <-+
j~j*(9c 0 9Ze) and
j*j*('R M OZC)
jj* (9
0 9zC) is
zero. We'll do only one of them (the second one can be done similarly). For the
first embedding the map a2 is zero (since we are projecting the second
JZC
onto 'Z)
whereas the first map (because of the relations in Indh(Zc)) is equal to minus the
composition
jj* (ac N 'R) -+ j~j*('R N 'R) 4 A,(-T) -+ A,(Ind (Rc))
which is exactly the third map when restricted to j*j*('Rc N 'R).
Recall that we defined 049Z) to be the quotient of Ind
h('Rc)
by the image of the
map Leib from 2.2.1 obtained using the combination 01-02-03. By construction we
have a short exact sequence
0
-+
''-
Qje(9)
->
Q('R(2)
-+
0,
(2.12)
where 'R' is a certain quotient of R. In the rest of this section we will show that the
above extension is in fact isomorphic to the extension of Q ('R) given in Theorem
2.1. This is equivalent to the following:
Proposition 2.2.2.
Consider the extension of Q1(')
have
= 'R.
1. Z'R
56
given by (2.12).
Then we
2. The pull-back of (2.12) via the differential d :9Z
Q 1('R) is the original
-+
sequence (2.10).
Proof. To show that 'R' = 'R, consider the chiral extension given by the equivalence
of Theorem 2.1.
This is an extension of Q1 ('R) such that the pull back via the
differential R -+ Q'('Z) is the sequence (2.10). that is, we have the following diagram
0
2
Qc(WR)
'R-
,
de
0
(2.13)
0,
d
>0
>
*r
>
:z
--
i(')
with dc a derivation, i.e. as maps from jj*('Rc MJZc) to A,(Oc('R)), we have dc(IIc)
pAJ"
=
a -(dc, ir), where pc is the chiral product on 'Rc and
(3)(7r, dc) - o- o
pzQc(T) is the chiral action of JZ on Qc('Z). We claim that there is a map of short
exact sequences
0
:At('R)
Ai (Indh (gc))
:A ('R 0R) -
0
,
0
jid4I
0
>
A('R)
>
that factors through 0 -+ Ai('R')
A(Q'c(9z)
-+
A-(4))
-
induces an isomorphism from p1 ('R) to Q 1('R).
(Q i('R))
A!(Q 1 ('R))
-+
0, and moreover
This would imply that JZ', which
a priori is a quotient of R, is in fact 'R itself. Furthermore, the fact that it is an
isomorphism on Q1 ('R), would also imply that Q0R)
via d : 'R -4
~ Qc('R), hence the pull-back
l('Z) would indeed be the original sequence 0 -9Z 4 'R -+ 'Z -+ 0.
To prove the claim, consider the map dc .
Zc
-
Qc('R) given by (2.13). Using the
chiral '-module structure pzac(T) on Qc('R), we can consider the composition
jj *('R X 'RC) iN
j~j*('RZM
57
c
9)
ci(
g))-
The above composition can be extended to a map from Ai (R) D
j,j*(T M
'C)
-
A!(Qc('R)), by setting the map to be Ai(i) on A,('R). It is straightforward to check
that this map factors through a map Dc
D': A, (Indh('Rc))
(
Note that, by construction, the resulting map Dc -Z'0
9'R 4 Q('R) is the one given
by z 0 w 4 zdw, for z and w in 'R, and that the kernel of this map is just the ideal
defining the Leibniz rule.
To show that De factors through 0 -
A!(J'')
-
A(Qc(T))
+
A!(
1 (R))
-+
0, we
need to show that the composition of De with the map
Leib: A,(RC
0
-+ Ai(Indc(Rc))
3zC)
given in 2.2.2. vanishes. Hence we are left with checking that the composition
A (C 0 TC) Lb Ai (Indh(ZC))
(Dc
c((,))
is zero. For this, recall that the map Leib was constructed using the linear combination ai - a 2 - a 3 of three maps ai, a 2 and a 3 from j~j*(Z'
JUc). By looking at
the map
01-a2-0 3
j'j*('RcZc)
Ai (Ind
AS(DC
c
'c))
we see that the condition on dc being a derivation, implies that the above composition vanishes. Indeed A (Dc) o ai is given by
jT j * (9ZCZ(DC) orad
give
by t
ao idbdc
a
y
the t9ra
C
o
variables
The map A! (Dc) 0 a2 is given by the above by applying the transposition of variables
58
o, whereas the third map
j~j*24
is equal to j~j*(9c M 9Zc)
-L-)4 A, (Rc) -+ A, (Indc (gc))
A, (c)
/C
i(c(:)).
d)
g('
Ai
Therefore the above maps
coincide with the terms in the relation dc(pc) = pgQc()(7r, dc)
-o.g2,Qc(j)oo-(dc, 7r),
and hence A, (Dc) o Leib in zero. Note that the resulting map
A (Indi(2)
A, (9Z (&J)
> A, (Q I(2))
induces an isomorphism
AI(OPM) ~-A,('R (&9)/Im(7r o Lei-b) ~
iG(R)
This conclude the proof of the proposition.
2.3
Quantization of the center
Recall the commutative chiral algebra
mentioned in the introduction,
3cit
3
3 cit
,it defined as the center of Acit. As we have
can be equipped with a chiral-Poisson structure
in the following two equivalent way:
* For any h
#
0 let K be any non critical level
Ah the chiral algebra A,.
1
= ncit
Let z and w be elements of
+
hAnill and denote by
3cit.
Let z" and w, be
any two families of elements in Ar, such that z = z, and w = w, when h = 0.
Define the Poisson bracket of z and w to be
{z, W} =
[z., W]^Ah
h
59
(mod h).
We will now introduce the Drinfeld-Sokolov reduction
"Ix
and explain how this
functor provides a 1-parameter family of chiral algebras {Wa} :
that Wo ~
2.3.1
{Tx(Ah)}
such
3rit, that are a quantization of 3,it.
The Drinfeld-Sokolov reduction
We start by recalling the BRST complex for a chiral algebra A as presented in [BD].
We will also recall the BRST reduction for A-modules over X. Finally, in 2.3.1 we
will define the Drifeld-Sokolov reduction "x
"Ix : Ah-mod
as a special case of the above,
-+
Wh-mod,
and use it to define the quantization {Wh} of
3
cit.
The BRST reduction
For a finite dimensional Lie algebra L, consider the Lie*-algebra 2:= L 0 'Dx over
X.
As it is explained in [BD], we can construct the Clifford Chiral algebra Cl'(2)
given by
Cl'(2) :U([]E
2*[-1] (DQx)'.
We can also consider the PBW-filtration on Cl'(2) (e.g. Cli(Z) = 2[1]e2*[-1]e9x)
and the adjoint action ad of 2 on itself. We define
2 Ta4e
to be the bull-back of the
following short exact sequence:
0
:
-x
>
el02 (2)
Cl<(2)/f2x ~g(2)
-a
0
>
x
>
1:2
-Tate -+ U(2-Tate)',
0.
ad
2
"ate
We have also an embedding i : 2[1]
>
0
Cli'(2). Consider now the natural map
where U(2-T4te)' is the chiral twisted enveloping algebra
60
of 2-Tate If we denote by A' the chiral algebra
U(2C-T*)'
& Cl'(2)
we get a map
Lie:= a+l : 2-+ A
given by g - g01+19a(g). If we denote by Zt the DG Lie*-algebra Cone(2 i
2)
we can view i and Lie as the components of a map of graded Lie*-algebras
4:
Ct -4
A'.
We now define a differential on A in the following way. As it is explained in [BD]
1.4.10, there is an action of £f on the DG-algebra Sym(2*[-1]) compatible with
the differential 6 of Sym(2*[-1]) (see [BD] 3.8.9) and one has the following:
Lemma 2.3.1.
The operations
[Lie, id2-[_1]], [i,
ol*-_11]
: 2 M 2*[-1] -+ A(Al)
coincide.
The above lemma allows us to define the map x. In fact it tells us that the map
p(Lie, id*[_1 ]) - p(i, 6|*.[.1]) : j*j*(2 M V*[-1]) -+ A,(A'[1])
vanishes on 2 M
*, where p is the chiral operation on A'. Hence we get a map
x : 2 @ V* -+A
A[1].
We have h(2
2*) ~ End(2). We define
Q to
be the image of the identity endo-
morphism of 2, projected onto h(A'[1]),
Q = X(Idc) E h(A[1]).
61
(2.14)
We have the following proposition:
Q is the unique element in h(Al[1]) such that [Q, i(g)] = Lie(g)
moreover [Q, Q] = 0. In particularif we denote by d the Lie action
Proposition 2.3.1.
for every g E Z,
of Q on A', then d is a derivation of A of degree 1 and square 0 and the map
# : Zt --+ (A', d)
is a map of DG Lie*-algebras.
The complex BRST(A) = (A, d) is called the BRST-reduction of U(2-Tate)'.
f
For a map of chiral algebras
: U(-Tate)'
-+
'R, the complex BRST(A:
'R 0 el'(2), dz := [f(Q), ]) is called the BRST reduction of 'R. The cohomology
of BRST(Az) is called the semiinfinite cohomology of 'R.
BRST reduction for modules
For an A'-chiral module M on X, the Lie-action QM of
Q
on it is a derivation
of square 0 and degree 1. If we denote such derivation by dM, then (M, dM) is a
BRST(A')-module. In particular we can take an U(2-Tte)'-module M and consider
the A'-module M 0 Cl' ().
f : U(2-Tate)'
-4
Moreover, if we are given a map of chiral algebras
'R we can do the same construction for any 'R-module M and get
a BRST(A,)-module. Hence we obtain a functor:
{'Z-modules on X} -- + {BRST(Ai)-modules on X}.
(2.15)
Drinfeld-Sokolov reduction
In the above framework, we can take L to be the nilpotent sub-algebra n of g. We
denote by Z, the corresponding Lie*-algebra.
62
Given any invariant bilinear form
r = rcit
+ hr'Kill on g, we see that the natural map
lifts to a map 2C--te -+ Ah, and therefore to a map
We can consider the complex BRST(An 0 el(92 )). This complex, comes equipped
with a differential d =
f(Q),
as explained above. However, to define the Drinfeld-
Sokolov reduction we will consider a new differential dn = d +do, where do is defined
in the following way.
Let {e,, a E TI} be a basis of n, and let Xo be the non-degenerate character of n
given by
=
Xo(e.)
1 if a is simple
0, otherwise.
As it is explained in [BD] 2.6.8.
this defines a map To : ,
regard as an element in 9* C Cl(Z,)
-+
-+
Qx that we can
Ah 9 Cl(2). We define do to be do := Xo, ].
Clearly d2 = 0, moreover do commutes with the differential d.
Definition 2.3.1. We define the Drinfeld-Sokolov reduction of Ah to be the DGchiral algebra
BRSTX (Ah)
(Ah & Cl( 2), dn = d + do).
We have the following remarkable theorem, proved in [FBI.
Theorem 2.2.
* For any
(Ah 0 Cl(2n), dn = d +do)
t,
=
,-it
+
h -Kill, the cohomology of the complex
is concentrated only in degree zero. Thus, this DG-
chiral algebra reduces to a plain chiral algebra Wh := H 0 ((Ah 0
d + do)) called the W-algebra.
63
l(92n),da
=
e The chiral algebra Wo is isomorphic to the center 3,it of Ait. Moreover this
gives a quantization of 3,it viewed as a chiral-Poissonalgebra.
From now on, we will denote by qIx the functor BRSTX. As we have seen in
2.15, this defines a functor
x : Ar,-modx:= {An-mod on X}
We will be interested in r, =
W-modx.
,.it. By the above theorem, for the critical level we
can re-write the above functor as
I x := BRTSX : Ac,.it-modx -+ 3c,.it-modx.
2.4
2.4.1
(2.16)
Main theorem
The chiral algebra of twisted differential operators on
the loop group
We will now recall the chiral algebra D, of r, twisted differential operators on the
loop group as presented in [AG|, for r, = r,-it +
hr'Kill-
Let G be an algebraic group and consider the group scheme G x X on X. Let Jx (G)
be the corresponding 'Dx-scheme, where, by Jx we denoted the functor
Jx : {Ox-schemes} -+ {Dx-schemes},
right adjoint to the forgetful functor
For: {'Dx-schemes} -+ {9x-schemes}.
64
(2.17)
Consider the Lie*-algebra L., and the invariant bilinear form r, = rit
The algebra 'Dr is constructed using the Lie*-algebra
OJx(G)
+
on g.
hKKi
( Lg. More precisely,
it is defined as
'Derit :=
U(0 Jx(G) (D Lg)/I,
where I is the ideal in U(OJx(c) D L.) generated by the kernel of the map
U((9Ja(c)) -+ Jx(G).
As it is explained in [AGI, the fiber ('Dr), of 'Dh at x E X is isomorphic to
0
('D). ~_U(g.)
(9 G[[tl].
U(g[[t]]®C)
Moreover 'Dh comes equipped with two embeddings
An 1 'Dr,
(2.18)
A-n
corresponding to left and right invariant vector fields on the loop group G((t)). In
particular, for h = 0, we have An = An = Acrit and 'Dr = 'Dit. Therefore we
obtain two different embeddings, 1:= lo and r := ro of Acit into D,it
Ac,.it I Derit
'
<'- Acrit.
If we restrict these two embeddings to
3,it, as it is explained in [FG] Theorem 5.4,
we have
(3 crit) = l(Acrit)
n r(Acit)
= r(3cit).
Moreover the two compositions
3
crit
"
Acrit
D C
65
'_
Acrit +-
3
crit
are intertwined by the automorphism 7 : 3era -+ 3,.a given by the involution of the
Dynkin diagram that sends a weight A to -wo(A)
(i.e. when restricted to 3cit we
have I = r o ).
The two embedding I and r of Ac,
structure of -era-bimodule.
into D,u endow the fiber ('Deru)2 with a
The fiber can therefore be decomposed according to
these actions as explained in the introduction.
These decompositions coincide up to the involution q and we have
('Dci-),
(13
('Dc,-),
A dominant
where ('D,.i)|is the direct summand supported on the formal completion of Spec(3Au).
Recall that we denote by '
the chiral algebra corresponding to ('Der)2. The em-
beddings 1 and r give 'Dc,. a structure of Ac,-bimodule, and we denote by 6C,.i the
resulting chiral algebra
CO.,.a := (Tx 0 Tx) ('Do,).
The following theorem relates the chiral algebra 'Do, to the extension of Ql( 3
rit)
arising from the quantization of 3 cit given by the 1-parameter deformation {Wa} =
{'Ix(Ah)} defined by 2.2, under the equivalence of theorem 2.1. This is the main
result of this chapter.
Theorem 2.3. The chiral envelope U(P(3c,t)) of the extension
0 -+ 3cr -+ c( 3 ert) -+ Q( 3 ,it) -+ 0,
given by the quantization {Wh := 'Ix(A)} of the center 3ert, is isomorphic to the
chiral algebra Co,.t.
66
2.4.2
Proof of the theorem
The proof of Theorem 2.3 will be organized as follows: we will give an alternative
formulation of the Theorem that consists in finding a map F from GV(3ci,) to
c0,. with some particular properties. The definition of the above map will rely on
the explicit construction of the chiral extension Qc(3crit) that was given in Section
2.2.2. In Section 2.4.3 we will finally define the map F and conclude the proof of
the Theorem.
Reformulation of the Theorem
We will show how to prove Theorem 2.3 assuming the existence of a map F from
Gc( 3 cit) to
COC,.t. In order to do so, we will use the fact that both U(Q2(3cit)) and
C,2 can be equipped with filtrations as explained below.
The chiral algebra U(Qc(3cit)), being the chiral envelope of the extension
0
-
3
crit -
f'(
3
crit) -+ Q 1 (3crit) -+
0,
has its standard Poincari-Birkhoff-Witt filtration. In fact, more generally, given a
chiral-extension ('R, C, Le, Z), using the notations from Definition 2.1.6, we can define
a PBW filtration on U(C, ZC) as the filtration generated by U(C, L-)o :=
U(C,, L)1 := Im(jj*(Lc
-
#c(C)
and
+" 'le
jj*(U(C, LC) Z U(C, Z')) - Ai(U(C, LC))).
Moreover, according to [BD] 3.9.11. we have the folloing.
Theorem 2.4. If'
and C are Ox flat and L is a flat 'R-module then we have an
67
isomorphism
e
Synl
-4 gr.U(C, Zc).
By applying the above to the case where C = 'R = 3,it and the extension of
Z = Q'(3crit) given by Z' = Qc(3,it) we get
gr.U(QC(3crit)) ~ Sym3*citQ(3erit).
The filtration on
c,-it is defined using the functor Tx from 2.3.1.
Recall that, for any central charge K = hskill +
crit,
the functor Tx assigns to
a chiral Ar,-module a qIx(Ah) = Wr,-module. In particular, for every chiral algebra
3, and every morphism of chiral algebras
f
#
Ah -+ 3 we have
chiral algebra morphism
chiral algebra morphism
4: A A-+ 3
Wa -+ x(B)
Jx(#):
Moreover recall that for h = 0 we have xIx(Ait) ~ 3,it.
As it is explained in [FG], the chiral algebra C,.it can be described as
(RxZ Tx) (U (C,Zc)) ~+ 6C,-it = (xfx Z Tx)('Do.,it),
for some particular chiral algebra e and chiral extension L'. Hence it carries a
canonical filtration induced by the PBW-filtration on U(C, L). We will recall below the definitions of the chiral algebra e and the chiral extension Le .
THE RENORMALIZED CHIRAL ALGEBROID. Recall that [FG] Proposition 4.5. shows
the existence of a chiral extension A"',' that fits into the following exact sequence
0 -+ (Ac.it 3 9 Acit ) -+ Aren,
crit
68
- G1(3cit ) -+ 0,
which is a chiral extension of (Ait3 0 A,i) in the sense we introduced in Definition
crit
2.1.5. In particular, if we consider the chiral envelope U((Acrit 3 9 Ac0it) Are"'r), by
crit
Theorem 2.4 we have
gr.(U((Acrit 3 0 Acrit), Aref"'))
crit
S(Acit 3 0 Acit) 3 0 Sym3.c (Q'(3crit)).
crit
crit
The chiral envelope U((Ait 0 Ait), Aren') is closely related to the chiral algebra
3crit
DOcrit'7in fact in [FG] the following is proved:
Theorem 2.5. We have an embedding G of the chiral extension ArenT into 'Dit
such that the maps 1 and r are the compositions of this embedding with the canonical
maps
Ac,.it -t (Ac,.it (2 Ac,.it) - A''
3crit
The embedding extends to a homomorphism of chiral algebras
U ((Ac,.it 3 0 Ac,.it), A r*"'' ) _4 'De,t
crit
and the latter is an isomorphism into 'D
Therefore we see that 6cit is given by applying the functor Tx N Tx to the chiral
envelope U(e, Zc), for
C = (Ait
0
Acit), and L'
-
A'.
3crit
In particular, since the functor qJx is exact, we obtain a filtration on C'cit induced
from the PBW-filtration on U(A,it
Ait,
Are"'j) such that
0
SymcitQ'(3c,.it) ~> gr.'3 ,
69
where we used the fact that qIx(Acrit) ~ 3 crit.
Remark 2.4.1. Note that if we apply the functor Tx to the two embeddings in
(2.18), we obtain two embeddings
Wh -+ ('Ix Z
such that 1 := l
;
-x)(Dr)
W-A
= ro or =: r o r/, where we are denoting simply by la and rA
the maps 'Ix(lh) and 'Ix(r) respectively. In particular, for h = 0, we obtain two
embeddings 1 and r of
3cit into ('lx N Tx)('Dc-it) that differs by r/. Moreover the
image of the two maps lands in 30, therefore we obtain two embeddings
3crit I+ 3 +
3 crit.
From the above construction it is clear that
3
,it corresponds to the 0-th part of
the filtration defined on 30. Moreover, by the definition of the map G from Theorem
2.5 (see [FG]), the embedding
3
crit
" 30 induced by the inclusion (Ait3 0 Ait)
-4
crit
U((Ac.
9 Acri), Aren'r) under 'Ix Z Tx, coincides with 1.
Suppose now that we are given a map F : Qc( 3 ,it) -+ C,-it satisfying the conditions stated in Definition 2.1.6. By the universal property of the chiral envelope,
we automatically get a map
U(Ge(3crit)) -+ Cc,-itClearly not every such map will induce an isomorphism between the two chiral
algebras.
Theorem 2.3 can be reformulated as saying that there exists a map as
above, that gives rise to an isomorphism U(Qc(3crit))
have the following:
70
~+COit. More precisely we
Theorem 2.6. There exists a map F : Ge(3,-it)
-+
(Ccit)1
-
Cct compatible with
the
3eit structure on both sides that restricts to the embedding 1 of chiral algebras
on
3cit such that the following diagram commutes
Q(3rit)/
3
crit
C it) 1/3crit.
F
Q 1 (3crit)
We will now show how Theorem 2.3 follows from Theorem 2.6. The proof Theorem
2.6 will occupy the rest of the article.
Proof of (Theorem 2.6
=-
Theorem 2.3). To prove Theorem 2.3 we need to show
that the above F induces an isomorphism U(Qc(3crit)) ~> Ccit. This amounts to
showing that the following diagram commutes for every i:
gri+1U(Qc(3crt)
Ei+1
F
crit'
(2'19)
Sym'+ g1'(3crit)
But this follows from the fact that the above filtrations are generated by their first
two terms. In fact, more generally, for any chiral envelope U(,c), we have
Ai(gri+1U(L c)) :=
Im
jj*(U(zc)
1
I
z U(LCC)i)
A! (U(LC))
Ijj*(U(Lfc)
/
1
z U(LC)i-i) -+
A I(U(fc))
It is not hard to see that the isomorphism Symi+ 1
71
Q'(3erit)
/
~> gri+1U(Q"(3crit))
(and similarly for C',.it) is the one induced by the map
j~j*(Q (3 crit) M Sym,) -+ A (gri+1 U(G(3crit))),
that in fact vanishes when restricted to Q1 (3erit) MSymi,
and factors through the
action of 3crit. Therefore the diagram (2.19) commutes by induction on Z.
2.4.3
5
Construction of the map F
Recall that Theorem 2.6 amounts to the construction of a map of Lie* algebras
F : Ge( 3 erit) -+ CO-it compatible with the 3cit-structure on both sides and such
that:
1. F restricts to the embedding 1 (given in Remark 2.4.1) on 3,it.
2. The following diagram commutes:
G(3crit)/3rit
Fcr
Q1(3crit)
Remark 2.4.2. Since A, (Qc(3eit)) was constructed as a quotient of A (Ind3 crit ( 3 c it))
and since, by definition,
At
(Ind3c,.(3c,.it))= A,(3crit) @®jj*(3c-it0
3 cit)/
3
crit M3 cit,
to construct any map F from Qc(3cit) to C. it we can proceed as follows:
. first we construct a map
f
: 3 cit
-
C.
72
Using the chiral bracket p' on C,
we consider the composition
j.j*3(3c,, Z
jj*(CO
i
o
4
,(Co
This composition yields a map
3
Fci
rt
A!
~
Y
#:A(3c,-e) (Dj~j *(3c,4
by sending A(3cit) to CO.,
it)
,(Oi)
-+
3c,;)Cr
(0,r)
via A,(l).
" We check that the above map factors through a map
A,(Ind 3 ri1(3c,-it)) -+ Al(60,.t).
* We check that in fact if factors through F: A (Ind h., (3 ,.))-
"
We verify that the relations defining
Ai(,.).
Ai(Qc(3,-it)) as a quotient of A (Ind .i(
3 .,it))
are satisfied, i.e. that F gives the desired map F from Q(3c,.i)C to CO..t under
the Kashiwara equivalence.
Remark 2.4.3. Note that any map F constructed as before, automatically satisfies
the first condition in 2.4.3, hence to prove Theorem 2.6, once the map
f
is defined,
we only have to verify that condition (2) in 2.4.3 is satisfied, i.e. that the diagram
above commutes.
Definition of the map
f.
We will now define the map
f : 3 ..
-+ C
,it and hence, according to the first two
points in 2.4.2, the map F : A,(3,.it) T jj*(3.t Z 3..t) -+ At((CO ,t).
that it factors through a map F : Q(3.i;)
satisfies the second condition in 2.4.3.
-+
C'.,
Assuming
we will then show that it
This will conclude the proof of Theorem
73
2.6. The poof that it factors through D"(3cit) (which amounts to the proof of the
remaining last three points in 2.4.2) will be postponed until 2.4.4. To define the
map
f : 3c.;
-+ C-it we will use the following three facts:
1. There exist two embeddings
3
crit
-
6.it &
3
crit
constructed by applying the functor Tx to the two embeddings in (2.18). In
fact, by doing it, we obtain two maps
Wal
such that 1 := 10 = ro o'
(Ix Z 'Px)('DA) 41 W-rj
=: r o 7, where we are denoting by lh and rh the
maps Tx (l,) and qPx (rh) respectively. The two embedding of
3
crit
correspond
to the above maps when h = 0.
2. There is a well defined map
e : Wr -4 W-a.
In fact, since Wh = Ix(Ar), and since Aqr is isomorphic to Ah as vector space
with the action of C[h] modified to h - a = -ha, a E A-h, we can consider the
map Wh -+ Wqa that simply sends h to -h.
3. The involution 7 :
3
cit
-
3eit can be extended to a map 7 : Wh -+ Wh by
setting 7(h) = h.
74
We define
f
in the following way: for every zh E
f (zh) =
3',.t =
1ln(zh) - r(r(~a)
h
This is a well defined element in
e0c,.t because
Wh/h 2 Wh we set
(mod h).
10 = ro or, i.e. the numerator vanishes
mod h.
Assuming the proposition below, we will now show that the resulting F satisfies
condition (2) of 2.4.3, which, according to Remark 2.4.3, concludes the proof of
Theorem 2.6. Proposition 2.4.1 will be proved later in 2.4.4.
Proposition 2.4.1. The map
F: A,(3crit) Djej*(3cru 0 3c,-) -+ A
obtained by using f : 3c,.i -+ C,.i from above, factors through a map F : A,(Qc(3ri))
A I(6c,.a).
End of the proof of Theorem 2.6
Proof. We are now ready to finish the proof of Theorem 2.6, which, according to
Remark 2.4.3, amounts to check that
Qc(3cric)r/3c
F
Or)1
/ 3 crit
Q'(3c,.t)
commutes. In order to do so, we will show that it commutes when composed with
the map d : 3cru -+ Q 1 (3crit). By looking at the composition
3 erit
A
Q1 (3c,.i) -+ Qc(3c-it)/3crt
75
4 (Co,-t) 1/3cru,
-
we see that, for z E 3ciit, the resulting map is
1 lh(zh)
2
-
where zh is any lifting of z to
3cit.
taking the quotient of C',it by
3,it.
rh(r(e(zh)))
h
(mod h),
(2.20)
Note that this map is well defined only after
For the other composition, we first need to recall how the isomorphism
1(3i2t4
(6,.t)1/3crit was constructed. Recall from 2.4.2 that the filtration on C,-it is the one
induced (under x1 x 0 xIx) from the isomorphism G given in Theorem 2.5. Therefore
the isomorphism above is the one corresponding to the composition
(Acrit 3 9 Acrit) 3 0 Q1 (3erit)
crit
crit
U(A*ren*)1/(Acrit 3 ® Aerit) G
crit
'Dcit/l(Acrit)+r(Acit)
under (xIxEPx) (here, for simplicity, we are denoting the chiral envelope U((Acrit 0
3
crit
Acrit), Are")
by U(Arn')). If we consider the inclusion of Ql(3crit) followed by
the first arrow from above, it is clear that the image in U(A'"')/(Ait 0
3
Acrit) is
crit
G[[t]] x G[[t]] invariant. In particular it means that the image of G 1(3cit) maps to
(x
X Fx)(U(Aren',))/3crit. Now, by looking at the definition of the map G (see
[FG] 5.5.), we see that the the map
3crit
+ Q1 (3crit) -+ (,x M Tx) (U(A'"')1/3crit (x1)(G)
fei)1/3crit,
O
is indeed given by (2.20). This completes the proof of Theorem 2.3.
O
We will now give the proof of Proposition 2.4.1, which will occupy the rest of
the chapter.
76
2.4.4
Proof of Proposition 2.4.1
Recall that the proof of Proposition 2.4.1 consists in showing the following:
1. the map
: A (3,it) D jj* (3cit Z 3c.it) -+ A (COit) factors through a map
F : Ai(Ind 3cril(3c it)) -+ AI(60,.4
2. the map F factors through F : A (Ind'hj
(3,))
-+
A!(C%,;).
3. The relations defining A! (Qc( 3 ,it)) as a quotient of A! (Indch., (3..t)) are satisfied, i.e. F gives the desired map F from Q(3,it)c to C,.it under the Kashiwara
equivalence.
For this we will need the following Lemma.
Lemma 2.4.1. The composition
Wrl X W-
> (WFx Z Tx) ('Dr) Z (T x Z Tx) ('Dr) ± A,((Wx M Wx) ('Dr))
is zero.
Proof. In [FG] Lemma 5.2 it is shown that the composition
Art Z A-r
> 'D Z 'Dr, -
Al('Dn)
is zero. In other words the two embeddings centralize each other. The Lemma then,
immediately follows by applying the functor (x
Z Tx.
Proof of (1).To prove that P factors through
F: Ai(Indjht(32,.;))
77
c.
we use Lemma 2.4.1. Recall that we defined f from
f
1 l(zh) - rh(7q(e(zh)))
f
(h
3'ru to
C',. to be
(mod h).
Because of the above Lemma, it is clear that, when we consider the inclusion
3
3ert M
crit '- j~j*(3cruitZ3rit) and the composition with the map to A,(C' ,t), the resulting
map factors as:
3 crit
N 3 rit
3 crit
Z 3 crit
2
2
jj*( 3 crit z 3
:
a
rit)
,
(eA,.)
(3 crit)
which implies that the map factors through a map F
A (Ind 3 cri (3c,.it)) - At((C,-i)-
Remark 2.4.4. Note that when we restrict the map f : 3, i
., to
3cr
3
because of the flip from h to -h in the definition of e, we simply obtain the inclusion
3 crit
4
ecrit'
Proof of (2). Now we want to check that the relations defining A,(IndYh(3 ,.))
as a quotient of A (Ind 3 c1 ( 3 .,-)) are satisfied, i.e. that F factors through a map
F : A,(Indh
(3,.))
A
(,.
First of all, recall that to pass from A,(Ind 3 crit(3cit)) to A (Indh
(3. it)) we took
the quotient by the image of the difference of two maps from A (3ci @ 3 crui) to
A,(Ind3 ct(35,-u)). The first map was given by
A,(3crit @3er.i)
m A!(3ert) -+ A!(Ind 3 crit(3eit)),
while the second map was induced by the composition
j~j*(3erit
Z 3cre)
)
jej*(3crit Z
78
crit)
-* A,!(Ind3cri (3c,-t)),7
(2.21)
which vanishes on
3c,.it M 3c,.; - j,j*(3 c,. M 3c,) . When we compose the map
(2.21) with F, we get 1 o m
=
However when we compose the second
o (1 M 1).
map with
j,j*(3c,.
3.,.it) E4 jj*(,
e M .,.i)
Ai (C.,;),
because of Remark 2.4.4, we see that this map corresponds to p' o (1 Z 1) hence the
difference of the images goes to zero under F.
Proof of (3). Now we are left with checking that F factors through
FT: A, (Indsct(3c,.it)) -+ Ai(Qc(3c,-i,)).
This will occupy the rest of the article. Recall that A (Qc('Z)) was given as a quotient
i)) by the map Leib. The Leibniz relation was given as the image
of A! (Indh i (3,.
of a map
A,(3,.it 0 3.,)
-+
A!(Indih (3c;))
and this map was the sum of three maps, ai, a 2 and a3, from
which vanished on
3c..t
3Z.,.t.
j j*(3. i
3,.;i)
Hence we want to check that the composition
A!(Ind
A,(3,.t 0 3..t)
(
vanishes. Instead of considering the map from A, (3c,.t 0
A
(
3c.,.)
we can consider the
three maps
j~j*(3,.it M3Yt)
Cr
Cr
a2
C
(
A2 (Ind 3crit (crit)
79
AI
(2.22)
(2.22)t)
and show that the composition F o (a 1
-
a 2 - a 3 ) is zero. Recall that the first map,
ai, was given by projecting onto j~j*(3,t Z
3
), i.e.
a'
j~j*(3'c,.it Z 3Xit)
where
#
>j~j
*(3c,.t z 3,. i)
>Ai
(Indjh it(3c,.;t)) ,
denotes the second component of the projection
A, (3c,.;t) E j~j*(3c,-it Z 3 it) -* A i(Ind chit(3c
it)).
The second map, a 2 , was given by o- o ai o -, and the third map a 3 was given by
the composition
jj(3cc4, Z 3 it) -4 A,(3 it)
C
-*
A(Ind~ma(5t)A,
A! (d 3 rrit (3ccrit))
A (Indh N(3 ,t)).
-
When we compose a 3 with the map F : A,(Ind'h(3,.i))
-+
A!('B), it is easy to
see that the unit axiom implies that the composition is equal to
j~j*(3c, it 3cc,.it) 4 Ai (3c,.it)
>rA,(60C,.it).
(2.23)
Now consider the chiral algebra (qJx Z Tx)('Dr) and denote by p'i its chiral operation.
Consider the map
f :W
-
(-+x Z
f(z)f ==h2za
I lh(z) - r(7(e(z)))
(i.e. we are not taking this element
x)('D)
W
x(D~)
(mod h)). It is clear that the three maps a'',
a2 and a' given by
j~j*(Wh
Z
Wh)
">j~j*((,PxM Fx)('Dh)M (WFx M Tx) ('DO)
80
Wh)
i~*W
--
-* j*j*(&4Px Z Tx(rj
> A,(('Ix M x) ('D)) -+ Ai (C,.),
j~j*(Wh Z Wh) 4 A(WA)
respectively, vanish on j~j*(h 2 (Wh
Z
-~4
(a'3)
-4
j~j*(Wa Z WA), in particular they
3.,.t)
to A, (60,t). Moreover the resulting
WA))
define well defined maps from j~j*(3,. i
maps coincide with c,
( 4 'x Z 'Ix)(Dh)
a2 and as composed with F. In fact, the first and the last
coincide by definition. For the second one, simply note that, modulo h, the map
rA o 77 o e equals 1.
By the above, to show that the combination of the three maps given in (2.22)
is zero, it is enough to check that the combination a' - a' - a" of the above three
maps vanishes.
Let us denote by a', a' and a' the maps from j~j*(WflZWa) to At (('x
0 Tx) (DA))
corresponding to a', a' and a' respectively (i.e. before taking the maps
(mod h)).
We will show that the combination c' - a' - a' is already zero.
Because (Tx N qIx)(DA) is h-torsion free, it is enough to show that the three maps
agree after multiplication by h. But now note that each of the maps
ha', ha', ha' E Hom(j~j*(Wa 0 WA), Al(('Fx 0
81
x)('D))),
is the sum of two terms, and the sum of the resulting six maps is zero. Indeed ha'
equals the sum of the following two maps:
j~j*((,Px Z 'x) ('D)) L: A I('D),
j,j*(Wh Z Wa))
j~j*(Wh
Z Wh) lagrojo
((F ZT)
(2.24)
(2.25)
AI('Dh).
D
On the other hand, the map ha' is given by the sum of the following
jj*(Wa
Z Wh)
-4>
AI(Wh)
j~j*(Wa 0 Wa)
(2.26)
AI('D),
(2.27)
A!('Dr).
Al0
It is clear that the map (2.24) equals minus the map (2.26). Similarly, the relation
p = --
o p o o guarantees that the two maps summing up to ha', given by
(xZ'x)(raooes)o
j~j*(Wa 0 Wh)jj
0o'0
,
)
and
j*j*(Wh
Z
Wh)
(rro?7oeorho1oe)ooj*j*((,x
Z Tx)(Dh))
fop'
cancel with the remaining maps (2.25) and (2.27) respectively.
Hence the composition ai - a 2 - a3 as a map from
j~j*(3, i
zero, i.e. the map F : Ai(Indih (31,.t))
factors as
Ai(Ind
it(3c,.
Ain3cr
-+
)
A (0,t)
-) A
Cr)t)
82
i
).
3, ) to
A (C,-it) is
3
By Kashiwara we obtain the desired map F : O( c,it)c 4 C it, and this concludes
the proof of Theorem 2.6.
83
84
Chapter 3
Localization theorem for the affine
Grassmannian
3.1
Factorization categories and factorization spaces
In this section we start by defining the appropriate categorical setting that will be
needed to formulate Conjecture 1.2.2. In 3.1.1-3.1.5 we recall the definition of the
Ran space and define the notion of factorization category, algebras in a factorization
category, factorization spaces and factorization groups.
We will then recall the
factorization description of chiral algebras.
In 3.2 we will define and study the factorization category A-mod
A-modI} of chiral A-modules.
=
{I
-
In particular, in 3.1.8, given a commutative chi-
ral algebra 3, we will describe the category 3-mod, of chiral-3-modules on XI
as the category of modules over a sheaf of topological associative algebras on X,
canonically attached to 3.
In 3.2 we will recall what it means for a group G to act on a category C, as explained in [FG3]. In Section 3.3 we will generalize the above to a factorization group
9 acting on a factorization category. The factorization category we will focus on
will be the factorization category A-mod of chiral A-modules. In particular, we will
85
study the factorization sub-category A-mod 9 consisting of strongly 9-equivariant
objects in A-mod, as defined in 3.3.1.
In 3.3.3, we will apply this to the chiral
algebra D,.t and the group 'Dx-scheme JG of Jets into G as defined in 3.1.7.
3.1.1
The Ran space
Let k be an algebraically closed field of characteristic zero, and denote by fSchaf f/k
the category of affine schemes of finite type over k. For a category 'D, denote by
Pshv('D) the category of pre-sheaves on it, i.e.
Pshv('D) :=
{ functors
'DP -+ Gpd.j,
where Gpdo, is the category of oo-groupoids. Fix a smooth curve X over k.
Definition 3.1.1. The Ran space on X is the functor of points
fSch a! f/k
>
Set
{F c Hom(S, X) : F is finite}.
S
The Ran space is more commonly defined as the colimit of the diagrams
Fin'.
> Pshv(f Schaf f/k),
where FinP, denotes the category of finite sets with surjections as morphisms, and
a surjection of finite sets
#
: I -* J maps to the corresponding diagonal embedding
XC-< XI.
Note that a k-point Spec(k) -+ Ran(X) of Ran(X) consists of a finite collection F
of points F C X(k).
In this perspective, we can think of Ran(X) as the moduli
space for finite subsets of X. More generally, for an affine scheme S, an S-point of
Ran(X) consists of a finite set of maps F = {f1,...,
f}
c Hom(S, X). We asso-
ciate to F the closed subscheme in Xs := X x S given by the union of the graphs
86
1f t C Xs.
We will be interested in collections of categories over finite powers of the curve
X satisfying certain properties. We will use the notion of tensor product of abelian
categories as introduced in [FG7] 17.1.
As it is explained in loc. cit. the tensor product of two abelian categories is an
abelian category satisfying a certain universal property. In particular it may not
exist. We will always work in the following framework, in which the tensor product
does exist.
Let A
4
A' be a homomorphism of commutative algebras. Consider the abelian
categories A-mod and A'-mod. The map
f
gives rise to a monoidal action of A-mod
on A'-mod, moreover the monoidal action A-mod x A'-mod
-+
A'-mod is right-exact
and commutes with direct sums. Assume now that we have an abelian category C
endowed with a monoidal action of A-mod such that the action map is right-exact
and commutes with direct sums. Under the above assumptions, as it is shown in
[FG7], the tensor product
A'-mod
0
A-mod
C,
exists. For instance, we can consider an abelian category C, with a map from A
to the center Z(C) of it. This map endows C with a monoidal action of A-mod,
satisfying the properties above. In this case the tensor product A'-mod
0
C can
A-mod
be described as follows. This is an abelian category whose objects are objects C in
C, endowed with an additional action of A', such that the two actions of A coincide,
where one action is the one coming from
f
: A -+ A', and the second is the one
coming from the map A -+ Z(C). Morphisms are morphisms C 1
-+
C 2 in C that
intertwine the A'-action.
The categories of interest can be described as categories over the space Ran(X).
87
More precisely, we have the following definition.
Definition 3.1.2. A category C over the Ran is the following data:
" For a finite set I we have an abelian category Cxi over XI, in other words, we
are given an action of the monoidal category QCoh(XI) on Cxi. We require
the monoidal action to be right-exact and to commute with direct sums.
" We have pairs of mutually adjoint functors
A*
for every surjection
#
: CxI
exJ
'
:
A'P,,
: I -* J.
" We require that the induced functor
QCoh(Xj)
X
@
Ax *Ide
+x
QCoh(XI)
be an equivalence (as categories acted on by QCoh(Xj)).
We define the category C to be
ifl
Given a map
f
Y-
Cxi.
- X' and a category C over Ran(X), we denote by
the category over Y' given as
I
- f*()yi := QCoh(YI)
9
QCoh(XI)
where QCoh(XI) acts on QCoh(Y') via the map
88
f '*.
e
f*(C)yi
Definition 3.1.3. A functor F between two categories C and D over Ran(X),
consists of a family of functors
F1 : Cxi
such that, for
#b: I
Dxi,
-+
-* J, we are given isomorphisms of functors A* o F, ~ F o A*,,
compatible with higher compositions.
3.1.2
Factorization categories
For every partition 7r: I
-
J of I, i.e. I = LJJEJIj, where Ij = 7r-1(j), consider the
open subset
j(I/J) : UCI/J)
where U(IJ) =
(X) E X jxj
-+ XI,
X3 if (i) A r(j)}.
Definition 3.1.4. A factorization category is a category
for every partition r: I -
e over
Ran(X) such that,
J of I, we are given equivalences
(j(I/J))*( exI) ~ C i
0r,- 0 x
...
U(,/-7)
,
(3.1)
compatible with refinements of partitions.
Example 3.1.1. The most obvious example of factorization category is given by
the category QCoh(Ran(X)) of quasi-coherent sheaves on Ran(X), given by
I
-4
QCoh(Ran(X))xi := QCoh(XI).
Definition 3.1.5. Given two factorization categories C and D, a functor F : C -+ D
is called a factorization functor if it is compatible with the equivalences in (3.1), in
the sense that
(F)|uIj)
~ (Fri 0
89
-
0 F.)|IU(,j).
3.1.3
factorization algebras
Let C be a factorization category over Ran(X). We are interested in chiral algebra
objects in C. In particular, we are interested is unital chiral algebras in the factorization category QCoh(Ran(X)). These correspond to chiral algebras as defined in
2.1.
Definition 3.1.6. A (chiral) algebra A in C is the assignment I -+ A, E Cxi, such
that
" For every surjection < : I -- J, we have an isomorphism A*(A
" For every partition 7r : I -*
1
) ~ Ai.
J of I, we are given an isomorphism
(j(I1J))*(A1) ~-A,, Z -.
- -~ A1|(1p>)
I
.
Definition 3.1.7. A unital chiral algebra over a curve X is an algebra in the factorization algebra QCoh(Ran(X)) that satisfies the following.
e (unit) There exist a global section 1 E Ax, called the unit, with the property
that for every local section
f
E A(U), U C X, the section 1f
E Ax2|U2_A(x),
(defined by the factorization isomorphism), extends across the diagonal, and
restricts to
f
E A ~ Ax2IA(x).
Remark 3.1.1. As it is explained in [BD], chiral algebras as defined in 2.1 are
the same as unital factorization algebras in QCoh(Ran(X)). In fact we have the
following proposition.
Proposition 3.1.1. There is an equivalence of categories
Unitalfactorization algebras
~
Unital chiral algebras},
in-+
90
:= Ax 0 Qx.
given by the assignment A -+ A
Given a factorization algebra A in a factorization category C, there is a notion of
modules over A. The category of A-modules will in fact be a factorization category,
defined as follows.
Definition 3.1.8. Given a factorization algebra A in C, the factorization category
A-mod(e) of A-modules in C is the factorization category defined by the assignment
I -+ A-mod(Cxi), where A-mod(Cxi) is the category whose objects are collections
MILK E
eXIUK, for every finite set K satisfying the following:
" For every surjection # : K
-w
K', we have an isomorphism
A*,(MIUK)
2
MILK'-
"
For every partition r : I u K
-w
K' of I U K, such that I C 7r- 1 (k') =: 7 for
some k' E K', we are given an isomorphism
(j(IUK/K'))*(MIUK)
where I
"
ui K
= I1
...
11
Z MyIU(IUK/KI)
A,
ui -.-.
U I,,_1 ui I.
The above isomorphisms are compatible with sequence of surjections I U K
->
K' "' K" in the obvious sense, where
I UK
= 7 U kEKI
' -(k)
and I
kjyk'
=
I
Uk"EK" Ik,
with I C I.
k"#1
For a chiral algebra A, we can consider the factorization category
A-mod := { unital A-modules in QCoh(Ran(X))}
These are A-modules M in QCoh(Ran(X)) such that
91
(3.2)
* (unit) If 1 E Ax denotes the unit section in A, then for every local section
f
E M(U), U c XI, the section 1 M f E
J3VlX{1uIU({)u/u)_A(xI), (defined by
the factorization isomorphism), extends across the diagonal, and restricts to
f
3.1.4
E M ~ Mx{*u|A(xI).
Chiral modules over X"
From proposition 3.1.1, it is natural to expect the factorization category A-mod from
(3.2) to have a different description in terms of the right 'Dx-module Ad. Given a
unital chiral algebra Ad, and a finite set I, we can in fact define the notion of chiral
Ac-modules on X' in the following way.
For any finite set K and embedding <
I
-+
K, consider the Il-dimensional
closed sub-scheme HO C XK given by the union of the diagonal sub-schemes
HO:=
U
x
=
<,7
for i E K 1 } C XK1 X XI,
7r:Ki-*I
where K = K 1 U I, and xi and yj are coordinates on XK1 and X' respectively.
Consider the following diagram
HO
XKi
X
XI
U+,
PP2
XI
X K1
where UO denotes the complement of HO inside XKi x XI. For a quasi-coherent
sheaf M, on X', denote by ['e(MI) the module over XK1 x X'
Fi(M) := ii*(p*(
given by
)).
Definition 3.1.9. A chiral A"-module M, on X, is a quiasi-coherent sheaf on X,
92
along with a map
ItI jj'*(Ad M MI) - riciu{*} MI)
such that the following is satisfied.
" (unit) The following diagram commutes:
j*j* (Ad Z MI)
j*j*(Qx M Ad )
id a Fpcu{*}(Ml)
'
]pcru{*}(MI)
* (Lie action) pP12}3
i,{2,3}
jj*
(Ad
I
/12,{1,3} = P1,{2,3}
= /4{1,3}~-1,{23}
Ad
where
M) -+ jj* (Ad Z Fcu{*}(M'))
)
-+FIcru{*,*}(M
0 U 1 2 , and
p/'1,2},3 : j~j*(Ad Z Ad Z JI') -
A 1 2 jj*(Ad Z MI) -
-+
A 12*(FIcru{*}(M')) -
Iccu{*,*}(3
.
As it is explained in [NR], we have an equivalence of categories between the
category A-mod, introduced in (3.2), and the category of chiral Ad-modules on X',
where Ad = Ax 9 Qx.
Proposition 3.1.2. For every I finite set, there is an equivalence of categories
A-mod,
~>
{ chiralAd -modules on X,
given by the assignment M 1 -+ M'
M, 0
?I0
.
Given a Ad-module MI over XI, the corresponding factorization module is constructed inductively. For instance, for the finite set I
93
U {*}, the module MVIul*} is
defined as the kernel of the chiral action
A jj* (Ad Z M') -+
Iclu{*}(M N
which is surjective thanks to the unit axiom. The inductive step, constructing the
modules MIUK for any finite set K, can be found in [NR] 3.5.
Now, let Ad and
3
d be two chiral algebras and
a morphism of chiral algebras.
Let A and 3 be the corresponding factorization
algebras. Consider the factorization category
3
e
= A-mod. Since the map
#
endows
d with a chiral Ad-module structure, and therefore makes it an Ad-module over
X (and 3, and Ad-module over X'), we see that, by proposition 3.1.2, 3 becomes
a factorization-algebra object in the category A-mod. In fact, for every I we take
(3)i E A-modxi to be 3B itself, and for every K we take ( 3 )IUK to be the object in
A-modxUK given by the inverse to the functor in 3.1.2. We can therefore consider
the factorization category 3-mod(A-mod) as defined in definition 3.1.8. We have
the following proposition.
Proposition 3.1.3. We have an equivalence of factorization categories
3-mod(A-mod) ~_3-mod.
Proof. For every finite set I, there is a tautologically defined functor
3-mod(A-mod) -+ 3-mod1 ,
and proposition 3.1.2 says that this functor is an equivalence. In fact, let us consider
an object M1 E 3-mod(A-mod1 ). By definition, this corresponds to a collection of
94
objects MILK E A-modIUK, satisfying the factorization property
(j(IN)')*(MIUJK) -- BI,
SI.-i 0
' '.
'
ITU(I/J)
I for some
for every partition 7r : I U K -* K' of I U K such that I C r-'(k')
k E K'. However, each MIUK, being an object of A-modIUK, corresponds to a
collection of objects
MIUKUJ
E A-modIUKUJ satisfies the factorization property
(j(I))*(MIUKUJ)
AI Z
Api_ Z MT
...
for every partition I, U -..U I _ 1 u T of I U K U J, such that I U K c
therefore think of the object MI as a collection of objects
MILJKLJJ
I. We will
satisfying the
factorization property for 93 with respect to the K finite set, and the factorization
property for A with respect to the finite set J.
Clearly, given an object MI E 3-mod(A-modI), the collection MIUK defines a 93module structure on MI. Hence we have the functor mentioned above. Conversely,
given an object NI E 3-modI, and hence a collection of objects
NIUK
E 3-modIUK,
we can construct the modules NIUKUJ in A-mOdIUKUJ using proposition 3.1.2 in
the following way. Using the object NIUK we can construct a chiral- 3c-module
structure on it. More precisely, consider the map <p
the corresponding stratification HO 4 X x XILK
I U K -+ I U K U
{*}, and
4- UO. Denote by N{*, the object
NIUKU{*}, and consider the Cousin complex for NII given by the above stratification,
0 -+ NII -+ jj*(93
NIUK)
-{*}0-
-+*
It is not hard to see that the unit axiom for 3 implies a natural isomorphism
i~i* (NJ,)[l] ~ F+(NILK), and that the second map in the above sequence gives rise
to a 1d action on the right-D-module N2ILJK
NIUK
- Therefore, given
the object NILJK, we obtain a 3d-module N'UK on XIUK. Using the map of chiral
95
algebras V) : Ad -
'B", we can regard it as a Ad-module on XI"K.
Now, using
proposition 3.1.2, we can construct the corresponding factorization A-module, i.e.
the collection 2N IUKUJ.
3.1.5
Factorization spaces
We will now introduce the notion of factorization space. In particular we will be
interested in the factorization space attached to a Dx-scheme
'.
For Dx-schemes of
the form 1 = Jx(Z), where Z is an affine (x-scheme, and Jx is the functor defined
in 2.17. The corresponding factorization space will be studied in more details in
3.1.3. In 3.1.7 we will focus on the case of Z = G, for an affine group-scheme G over
X.
Definition 3.1.10. A factorization space
set, and
i
is the assignment I
-
'1,
for I finite
a space over XI, such that
" For every surjection <: I -w*J, we are given isomorphisms
" For every partition
~ j
ir : I -+ J of I, we are given isomorphisms
(j(I/J))(
A factorization space
'j|xj
1)
-
IlX ...
X
%InhU(I/J)
is called co-unital if it comes with a collection of maps
iI -4
X" X
J2
for each partition I = I1 LJ I2 which extends the corresponding map over the complement of the diagonal. We demand that these be isomorphisms over the formal
neighborhood of the diagonal.
96
Example 3.1.2. We will see later that the space of opers on X , as defined in [BD2]
can be organized into a co-unital factorization space Op.. Moreover, we will have a
sequence of inclusions
Op,
c
C Op
O
where the last two (non-counital) factorization spaces Opnr and Op' generalize the
notion of unramified opers and opers on the punctured disc as introduced in [FG3].
Remark 3.1.2. Given a factorization space
,
we can define the factorization cat-
egory QCoh( ) of quasi-coherent sheaves on it, by the assignment
I - QCoh( h).
In the special case in which we take 'j to be X1 , we recover the factorization category
QCoh(Ran(X)) from example 3.1.1.
3.1.6
From 'Dx-schemes to factorization spaces
Given a 'Dx-scheme
J1' } and M
',
we will now define and study factorization spaces J
= {I -+
= {I -+ M 1 } canonically attached to 1. In studying these spaces,
we will use the notions of ind-scheme and formal schemes. In particular, given a
map
#
:S
'
X' we will consider the union
X x S, and the completion
we can regard
UiEIII,i
UiEIFP,
of X x S along
UiEIIrr
of the graphs Fi's inside
UiEIP4i.
As it is explained below,
as a scheme or as a formal scheme. When regarded as a
scheme, we will denote it by Do and by D' the complement of F, inside Do. We
start by recalling the definition of ind-scheme.
Definition 3.1.11. An ind-scheme X is a presheaf on the category of affine schemes,
that can be represented by a filtered family of schemes,
X = l'
a
97
Xa,
where the transition maps i,,
- X# are closed embeddings, and the limit is
: X,
taken inside the category Pshv(fSchaff/k) as defined in 3.1.1.
We say that an ind-scheme X is of ind-finite type if each scheme X, is.
By a formal scheme we mean an ind-scheme whose reduced part is a scheme.
Let now S be a space over XI, we will now explain the difference between the formal
scheme Ui174 and the scheme Dp. Assume for a moment that X is the affine line,
S is just a point, and I is the one element set. Let x E X be the point corresponding
to pt
-+
X. Denote by
b2
the formal scheme
b2
:= lin Spec(C[x]/z").
n
We could have also considered the disc as a non-formal scheme, in other words we
could have considered the scheme D2
D2 := Spec(Um C[X]/z").
n
By definition of direct/inverse limit, for an affine scheme S, we have
Hom(Db, S) ~ Hom(Dx, S).
However, the formal disc Dx is "too small", in the sense that it makes no sense to
talk about the formal punctured disc D2 - x. The same is not true if we consider
the disc D,. In fact, the latter, as a scheme, contains the closed point x, and we
denote by D' the complement of x inside D2.
To pass to the general situation, if S is an affine scheme mapping to XI, we can
consider the formal-scheme
UiEIFk .
We write it as
UiEIIO = li
Spec(R/In),
n
98
where R is a topological algebra, whose topology is given by Is's. As before, this
formal-scheme is not good enough for us. In particular it doesn't contain
UiEIL'e,
as
a closed sub-scheme. Therefore it makes more sense to consider the scheme DO
(3.3)
Do := Spec(Um R/I,).
n
The scheme D, has a closed sub-scheme UiEII>,i, and we can consider the complement
(3.4)
D := Do - UiEIrI>,.
The factorization space attached to a 'Dx-scheme
Let ' be a 'Dx-scheme over X.
In the case that J is affine, this is the same as a
'Dx-algebra, hence, as we saw in 2.2, the same as a commutative chiral algebra. In
the general case, given a 'Dx-space
,
we can construct a factorization space J
as
follows. For every finite set I, and test scheme S, we define J1 i (S) to be
S (''''>X
,
J1 ( S) =
a horizontal section D,p -+
where by horizontal we mean a map of 'Dx-schemes. The above construction defines
a functor
{ Dx-spaces} -+ { Factorization spaces }.
The factorization space JJ naturally sits inside the factorization space M
defined
as follows. For every finite set I and test scheme S, we define M 1 (S) to be
(3.5)
MA :=
a horizontal section D-
99
where D' is the scheme defined in 3.4. Clearly Ji (S) corresponds to those sections
that are regular along UiElPi.
Remark 3.1.3. Given a commutative chiral algebra 2d, we can consider the corresponding 'Dx-algebra 3
:=
B' 9
* . Since 3 corresponds to an affine Dx-scheme,
we can construct the corresponding factorization space. It is easy to see that the unit
axiom on B" translate into the co-unital property of the corresponding factorization
space. Therefore we have a functor
{ Commutative chiral
3.1.7
algebras} -+
{
affine counital factorization sapces }.
Factorization groups
Let 9x be an affine 'Dx-group scheme. From the above construction, we obtain a
factorization group 'Dx-scheme, i.e. for each power of the curve X', we have an
affine group scheme
9
, := JI9x over XI.
Consider now a group scheme G on X, and let Jx(G) be the corresponding 'Dx-scheme, where Jx is the functor defined in (2.17).
JG the factorization space JJx (G).
Jx : {0-schemes} -+
Denote simply by
In this case, by the adjunction property of
{Dx-schemes }, we have
JG1 (S)
=
S
'
XI
a section Do -+ G
We denote by MG 1 the ind-scheme of meromorphic jets MG:
MG1 (S) =
S
(0"'.'
> XI
a section D'
We will mostly focus on the quotient MG 1 /JG.
-
M 1 Jx(G),
(3.6)
G
We will see later, that its closed
points can be described as the set of G-bundles on X with a given trivialization
100
outside a finite set of points.
3.1.8
Let now
Chiral modules over a commutative chiral algebra
3
d
be a commutative chiral algebra. In this section, we will see how we can
describe the category of 3-modules on X, as quasi-coherent sheaves over the space
MI 1 3 defined in 3.5, for the 'Dx-scheme
%3
corresponding to 3. More generally, in
3.1.1for a chiral algebra 3, not necessarily commutative, we will construct a sheaf
of topological algebras 3(I) over XI, such that modules over it will be equivalent to
the category of chiral-3-modules on XI. For this we will first need to recall some
constructions regarding the chiral envelope U(L) of a Lie*-algebra L.
For a Lie*-algebra L, we will start by recalling the definitions of Lie*-modules
and chiral L-modules over XI.
Definition 3.1.12.
e A Lie*-L-module MI on X' is a quasi-coherent sheaf on
X' along with a map
PI : L Z M1 -+ rcI1*}
(M)
such that the following is satisfied.
-
(Lie action) p41,2},3
1{23
[12,{1
=
o
=
1{,3
p/'1,2},3 = A I
=
: L NL Z M
}
-*2,
where
2,13} -
-+
L Z Fcru{l}(M)
c
,
and
PL : L Z L N M -+ A.(L) Z M -+ A.(FIcru{}(M)) --+ FIcu{*,*}(M).
* A chiral L-module MI on X' is quasi-coherent sheaf on X' along with a map
pL : j.j*(L Z MI) -+ Fjcru{ }(M)
101
such that the following is satisfied.
- (Lie action) Let U'
{y1 = xi} and
{y2
c X
= xj-. Denote by
We require that pii,2}3 =
p
1.,
,3}
: ji,{2,3}
j*(L Z L Z M) -+
P2,{1,3} = /11,{2,3} 0 0*2,
/1i,2},3
be the complement of the diagonals
x X'
j'*(L E L E M)
-
j'
the inclusion
/4{23}
j~j*(L
j'
: U' -+ X
x X'.
where
M L Z M) -+
and
A 12.jj*(L M M)
--
-
-+ A 12 *( FIcru{}(M)) -+
1FIcru{*,*}(M).
Unlike in the world of usual Lie algebras, for a Lie*-algebra L, the category of
Lie*-modules on X' is not equivalent to the category of chiral U(L)-modules on XI.
Chiral U(L)-modules are in fact equivalent to chiral-L-modules, in the sense that
there exist an induction functor Ind, establishing an equivalence of categories
Ind,: {Chiral L-modules on XI} ~+{ chiral U(L)-modules on X'}.
(3.7)
We can also describe chiral and Lie*-L-modules over X' as modules for some particular sheaves of topological Lie algebras on XI. For this, consider the following
diagram
H(1C10{I1
X X X,
U.
XI
X
Given a Lie*-algebra L, let
2(I) := hr(j*j*(p1(L)[-I])),
f(Il := hr(p1(L)[-I]) and
102
(3.8)
where, for a sheaf M on X x X', we define hr(M) to be
hr(M) := 4mp 2 ,(M/M ), M such that M/M is supported on H.
The objects in (3.8) are sheaves of topological (xi-modules, moreover they have a
structure of Lie algebra, coming from the Lie*-algebra structure on L.
As it is explained in [NR], we have the following proposition.
Proposition 3.1.4. Let L be a Lie*-algebra. Then the category of Lie*-modules
(resp. chiral L-modules) on X' is equivalent to the category of 2('>-modules (resp.
2(M)-modules).
Chiral algebras and topological algebras attached to them
Let '3
be a chiral algebra. We will use the previous subsection to describe chiral
'B-modules over X' as modules over a sheaf of topological associative algebras.
The ideas is the following: consider a classical associative algebra B, and denote by
BLie the corresponding Lie algebra. We have an obvious forgetful functor from the
category of B-modules to the category of chiral-BLie-modules. Since chiral-BLiemodules are the same as U(BLie)-modules we therefore have a functor
B-mod
-4
U(BLie)-mod.
Let now K be the kernel of the natural map U(BLie)
-4
B. It is clear that the
functor above defines an equivalence
B-mod ~ U(BLie)/K-mod.
We can apply the same idea to the world of chiral algebras. When the chiral algebra '3 d is commutative, we can furthermore describe the (commutative) algebra
103
corresponding to U(B"ie)/K as some scheme over X.
Consider the (commutative)-Lie*-algebra 3 Lie corresponding to Bc. Denote by
K the ideal in U(93Lie) generated by the kernel of the natural map
.
U(3Lie) -+ Bc
3
From the definition it is clear that we have the following lemma.
Lemma 3.1.1. The
Ind:
{
functor Ind' from (3.7) induces an equivalence of categories
}
Chiral 93d-modules on X'
~
{
chiral-U(3Lie)/ K-modules on X'
}.
(3.9)
Let's now consider the sheaf of topological (commutative) Lie-algebras defined
in (3.8) for L =
3
Lie. Denote them simply by 3'BI) and 3(I) respectively. Because
of the chiral algebra structure on
3
"1,we have maps
U(p( )) + 9(I).
Denote by KCM) the ideal generated by the kernel of the above maps. Consider the
equivalence of proposition 3.1.4
{
chiral-U(3Lie)-modules on X1
}
~ {U(3(,))-modules on X1}.
We have the following proposition.
Proposition 3.1.5. The composition oJ the above
functor with (3.9) induces an
equivalence
{
Chiral
3de-modules
on X'}
modules for the topological
4
associative algebra U(3(I))/K(')
104
Proof. Clearly the functor
{
3d-modules on X'} -+
{
U(3('))-modules
}
given by
proposition 3.1.4 factors through
{
-+
modules for the topological associative algebra U(3(I))/Kf)}
{
U(3())-modules
On the other hand, given a module MI for the associative algebra U(3(I))/K('),
we can consider it as a Lie-3(')-module. By the equivalence (3.7), together with
proposition 3.1.4, we can consider the module Ind'(Mr) as a chiral U(3Lie)-module
over XI, However, the fact that M_ was in fact a module for 3(I) when regarded
as an algebra, implies that the action of U(3 lie) on Ind'(MI) factors through the
chiral algebra U(3Lie)/K.
Now, by Lemma 3.1.1 we therefore have that MI itself
is in fact a 93C-module on XI.
Example 3.1.3. The commutative case:
denote by
B
For a commutative chiral algebra
3
d,
the corresponding co-unital affine factorization 'Dx-space. Recall from
3.1.6 and 3.5 that we have constructed spaces JAB and M,'W
over X'. Denote by
p, the natural map
pI : M, 1A
and by Me'
the object 0
Sym(3(I))/K(')
1
-
XI,
:= pI(O1y ,).
Note that we have isomorphisms
~ 3(i) (as sheaves) over XI. For I =
{*}, the fiber of 3({*}) at
any point x E X coincides, by construction, with the topological associative algebra
3ass
= Im
$3)s"introduced in [BD] 3.6. As it is explained in loc. cit 2.4.7, for this
topological algebra, the ind-scheme Spf(3ss) :=
Spec($a') is the space of
horizontal sections of Spec(3) over the formal punctured disc D*. In other words,
we have
Lwa i tnm ( ( r ea
=
Let now I be a finite set with ni elements, and T = (X1 , .
105
,Xn)
be a point in X',
}
with xi
$
(,B(I)),
xj for all i and
,!as
j.
Since both
(I)
and MI% factorize, we have
)&
N
0(re^as
..
(rl
and therefore we have
(orel
B(I)
By the above, proposition 3.1.5 can be re-formulated in this particular case in the
following way.
Corollary 3.1.1. For a chiral algebra '3
Chiral 93cmodules on X'}
we have the following equivalence
f4
continuous modules for the sheaf of topological
associative algebras (9d
In the special case in which the commutative chiral algebra
3
d
comes as the
'Dx-algebra corresponding to a Ox-algebra under the functor Jx from (2.17), we
will simply write JZ and MZ for the spaces JSJx(z) and MINJx(z) respectively.
Note that, by the adjunction property of Jx we have
MZ 1 (S) :=
S
X
a section D -+ M
}
and JZ is the subfunctor consisting of those sections that are well defined on
UiE1I>.
{
Therefore in this case we take 'B1 to be 0 Jx(z) and we have an equivalence
Chiral (DJx(z)-modules on XI} -4
continuous modules for the topological
associative algebra (9"'z
106
J
3.2
Action of a group scheme on a category
Let C be an abelian category, and let G be a group scheme. A weak action of G on
C consists of functors:
act* : QCoh(S) 9 C -+ QCoh(G x S) 0 C,
functorial in S
-+
G, and two functorial isomorphisms related to these functors.
" (unit) The first isomorphism is between the identity functor in C and the
composition
C
t
QCoh(G) o C -+ C,
where the second arrow corresponds to the restriction to 1 E G.
" (associativity constrain) The second isomorphism is between the two functors
C -+ QCoh(G x G) 0 C given by the following diagram
a"t>
QCoh(G) 0 C
4act*
I act,
x G) 9 C.
QCoh(G) 9 G "QCoh(G
Example 3.2.1. The tautological map
triv* :C -+ QCoh(G)
C, C
-+
G0
C,
defines an action of G on C. We will refer to this action as the trivial G-action on
C.
Definition 3.2.1. We say that an element C
107
E C is
weakly equivariant, if it comes
equipped with an isomorphism
act*(C) ~ triv*(C)
(3.10)
which is compatible with the associativity constraint of the G-action on C.
We denote by CwG the category consisting of weakly G-equivariant objects.
Example 3.2.2. If we take C = Vect, with the trivial G-action, then we have
Vecw,G = Rep(G).
3.2.1
Strong action on C
For a group scheme (or ind-scheme) G, we let G(1) = Spf(C E c g*) denote the
first infinitesimal neighborhood of the unit 1 E G, and we let G 1 be the formal
completion of G at the unit. A weak action of G on C is called strong if either of
the following equivalent conditions are satised:
" We are given have functorial isomorphisms between the functors act* for any
pair of innitesimally close points
#,#': S -+
G, satisfying certain compatibil-
ity conditions.
" We are given functorial trivializations of act* for any S
-+
G 1 , respecting the
unit, the multiplication, and the adjoint action of G on G 1 .
Remark 3.2.1. The second condition above is actually equivalent to a weaker
version. It is enough to be given a trivialization not on G1, but on G(1):
act*IG(1)
triv*IG(i)
which is compatible with the unit and the Lie algebra structure.
108
(3.11)
Definition 3.2.2. Given a category C with a strong action of G, an object in
C E CS,G is called strongly equivariant,if the two isomorphisms
act*(C) ~ triv*(C)
coming from (3.10) and (3.11) coincide.
We denote by
3.2.2
CG
the category consisting of strongly equivariant objects.
The case C = A-mod.
Let us consider a scheme (or ind-scheme as defined in chapter ??) X and a group
scheme G acting on it. Consider the map
act: G x X -+ X.
This defines a functor act* : QCoh(X)
-+
QCoh(G) 0 QCoh(X), and it defines an
action of G on QCoh(X) in the above sense. We have also the projection triv
G x X
-+
X, and we can consider the diagram
act
G xX
triv
:X
In this case, the objects of QCoh(X)wG are exactly those modules over X whose
pull-back on G x X along the above two maps are isomorphic. Obviously we have
a functor
QCoh(X)wG -+ QCoh(X/G).
The example 3.2.2 corresponds to the case X = Spec(k).
Let now X be affine, X = Spec(A), and suppose that A is acted on by a group
G. Then we have an action of G on the category of A-modules, that corresponds to
the above action on QCoh(X).
109
More generally, let A be an associative algebra acted on by G. This defines a
weak action of G on the category of A-modules. In fact, we have a map
G x A-mod
given by (g, M)
-+
-+
A-mod
gM, where gM ~ M as vector spaces, but the action of A on it
is defined by a.(gm) = ((g.a)m). This defines the required map
A-mod -
QCohG D A-mod, M
-4 M,
where M(g) = gM E A-mod. The objects of (A-mod)wG are those A-modules M
that are endowed with an action of G.
As we have seen earlier, given a scheme (or an ind-scheme X) acted on by G,
we have a weak-action of G on the category QCoh(X). However, by considering the
category of D-modules on it, we see how this category carries a strong action of
G. The weakly equivariant objects in 'Dx-mod are exactly the weakly equivariant
'Dx-modules, and the strongly equivariant objects are the same as D-modules on
the quotient X/G.
As before, if we take an affine scheme X = Spec(A), we can translate what it means
for the G-action to be strong in terms of the G-action on A. In this case, a strong
action on A-modules, translates into the existence of a map
g -+ A
that coincides with the derived action of g on A coming from the G-action on it.
More generally, let A be any associative algebra, acted on by G via a map G
-
Aut(A). The action of G on the category of A-modules is strong if the derivative of
the above map g -+ Der(A) factors through the algebra if inner derivations via a
110
G-equivariant map
#,
g
>
Der(A).
A
The objects of (A-mod) G are those A-modules M for which the following diagram
commutes
d(actc)
act
I
where actG : G -+ Aut(M) is the G-action of G on M coming from the forgetful
functor
(A-mod)
3.3
w,G
-+
wet,G
~ Rep (G).
Action of a factorization group on a factorization category
We now want to explain what it means for a factorization category to be acted on
by a factorization group S.
Let X be a smooth curve over k, and let C be a factorization category and
9
= {I
-4
9} a factorization group (see 3.1.7).
Definition 3.3.1. By a weak action of
act*: xi
-+
9 on C we mean a collection of functors
QCoh(91 )
9
QCoh(XI)
Cxi,
for every finite set I, compatible with the factorization isomorphisms. These functors
should satisfy the following:
e (unit) The first isomorphism is between the identity functor in Cxi and the
111
composition
act*
Cx1 --
; QCoh(91)
0
QCoh(XI)
Cx1 -+ Cx1,
where the second arrow corresponds to the restriction to the composition with
the pull-back along the identity section X' -+ 91.
e (associativity costraint) The second isomorphism is between the two functors
Cx,
-+
QCoh(9, x 91) 0 Cx, given by the following diagram
I;P
QCoh(9i)
Cxr
0
Cxi
QCoh(XI)
act*
QCoh(9I)
0
Itc;g1
i
-X
9
QCoh(S x 91) 0 Cxi.
QCoh(XI)
Example 3.3.1. For every I, the tautological map
triv*: Xi -+
QCoh(9,)
0
CxJ,
QCoh(XI)
C
F->
('9, 0 C,
defines an action of 9 on C. We will refer to this action as the trivial 9-action on C.
Definition 3.3.2. Let
group
e
be a factorization category acted on by a factorization
9. We say that an object M E C is weakly equivariant, if for every I we have
act*(M) ~:_
triv*(M).
We will be interested in the case of a group 'Dx-scheme acting on a chiral algebra
A. Although we can define what it means for an action of a factorization group to
be strong, we will spell out the definition only in the case C = A-mod.
112
3.3.1
Action on the category A-mod
Let A be a chiral algebra, and let 9 x be a group Dx-scheme. Denote by
9 the cor-
responding factorization group. We want to apply the discussion above to the case
of C = A-mod. For this, we first need to define what it means for the 'Dx-scheme
9 x to act on the chiral algebra A.
Given a group 'Dx-scheme 9 x, we consider its coordinate ring 09
as a commu-
tative chiral algebra endowed with a map
6 : 0 SX -+ Ogx 0 Oq
Ox
of chiral algebras, i.e such that the following diagram commutes
ii*(OS9 0@
i*i*(09
O
,(09s)
9)
I
I
0 cJ9 )
Z90
>
A,(0 9X 0(99k).
Definition 3.3.3. An action of a group Dx-scheme Sx on a chiral algebra A, is a
'Dx-map of chiral algebras
A
act
4OA009O,,
Ox
such that (act 0 id) o act = (id 0 6) o act.
The condition on act to be a map of chiral algebras, translates into the commutativity of the following diagram:
j~j*(A 0 A)
A,(A)
I
jj*(A @9 QZ
NI
A 9 09x)
>
,A 0 09x).
In this case the category A-modxi is acted on by the factorization group
113
9 attached
to 9 x, in other words, we have functors
A-modxi -+ A-modxi
QCoh(91 ).
0
QCoh(XI)
To see this, note that the action of 9 x on A defines an automorphism V;"' of the
chiral algebra A 0 (9.
act
This automorphism is defines as the composition
: A @9
9,
actoia)
A
9
idq
where m denotes the commutative product on (9q
#yaw
chiral algebra on X. We denote by
corresponding to #ac.
>A
9 0,
when regarded as a commutative
the map $yCt : A 0 (99, -+ A0
(909
Moreover, we have an obvious forgetful functor
99x-modxi -+ QCoh(9 1 ).
We define the functor A-modxi -+ A-modxi
QCoh(91 ) in the following way.
0
QCoh(XI)
To a module M1 over X, we assign the image under the above forgetful functor of
the object M1 E A-modxi
every finite set K, we take
0
9x-modx, defined in the following way. For
QCoh(XI)
(MI)K
to be MILJK 0(SIUK and for every partition
7r : I Li K -* K' of I L K, such that I C Gr-'(k')
7 for some k E K', we define
factorization isomorphisms as the composition
=
(i(I/J))*(M~IU)
A, Z.ZA
,,
0&
(j(I/J))*(-MIJ
.A
Mr( 0(%
1 Z.Z
1 9
-
OSIUK)
Z (M®
9 9T
id
__cN..,Oc
1
SA,,
Z
'''
0-1
(9 OsST
q '_1 Z MT0
M1 0
..
A,,,-1 Z MT
g90
114
'''Z..
Z091,- _Z
U
097 IU(I/J)
-
-*(/
where I U K = I L
...
LI -I,_1
U I, and where the first isomorphisms is the one
coming from the structure of A-module on X' on Mi.
The objects in (A-modxi)"' 9 1 are those chiral modules MI endowed with an
action of the group Vx-scheme 91.
3.3.2
Strong action on the category A-mod
We now want to understand what it means to have a strong action of 9 on A-mod.
For this, we will need some preliminaries concerning the notion of Lie'-coalgebras.
Definition 3.3.4. A Lie'-coalgebra on X is a 'Dx-module k on X endowed with a
map 6: 2 -+ 2 0 Z satisfying
a) (Id+r)o6=O
b) (Id+v+v2 )o(id+6)o6=0
where r(v 0 w) =w 0v and v(v 0w 0u) =
0 u0v.
Remark 3.3.1. Consider now Z = HomDe (k, Dx 0 Qx). This Dx-module has a
structure of a Lie*-algebra. In fact, more generally, for any Dx-modules M, N, V,
having a map
M-4NoV
is the same as having a map
MZ
-4 A,(V)
and iterating the process again gives a map M E N 0 V
-+
A,(Qx). This game
allows as to construct a map Z 0 Z -+ A,(L) from the original one. The Jacobi
115
identity follows easily from conditions a) and b). Hence we obtain that 2 is a Lie*
algebra if and only if Z is a Lie!-coalgebra. Note also that in the same way we can
show that for any k-module, the map
M-
M &k
(3.12)
gives us a map M M 2 -+ A,(M) which defines an action of 2 on M.
Consider now a group 'Dx-scheme
9x and consider the unit section X
+ 9 x.
As it is explained in [NR], £9x := u*(Q9x/x) has a natural structure of a Lie!coalgebra. In fact this is very similar to the fact that for an algebraic group G,
QG
has a structure of a Lie coalgebra. As we have said before, we can now consider the
Lie*-algebra
29x = Hom 9 x(2x'Dx
&Qx).
More generally, we can consider the group schemes
back along the identity section X'
-"'
Si over X', and take the pull
9, of the sheaf of differentials of 9j. We
denote such pull back by ks,
kg, := *(QS,711).
In this case the 'Dxi-module ks, acquires a structure of coalgebra over XI. Similarly
to what we have seen before, a module M, for 29, naturally becomes a comodule
for the coalgebra
9,,
{
29,-modules
}
++
{
2g,-comodules
Let now 29x be the Lie*-algebra defined earlier, and let 2
}.
(3.13)
be the sheaf of topolog-
ical Lie algebras defined in 3.8. Following [NR] , we have the following proposition.
Proposition 3.3.1. Let
9
x be an affine Dx-group scheme with Lie*-algebra 29x =
116
Homs. (1S., 'Dx 0 Qx). Then the Lie coalgebra £g, is isomorphic to
Now note that, given a factorization group
9,
a module M, for
9
, over X'
naturally becomes a comodule for Pg,. In fact the coaction map is obtained in the
same way as in the case of X = pt (in which case t9x = g*), i.e. you consider the
composition
M -+ M 0 CG -+ M @9 (G/(e)2
where (e) is the maximal ideal in
0
_+ M @g
G corresponding to the identity element.
In
particular, by proposition 3.1.4, and the equivalence (3.13), the we have the following
corollary.
Corollary 3.3.1. For a 'Dx-groupscheme 9 x, there is a functor
{9I-modules} -+ { Lie*-2x-modules on XI}
Consider now a chiral algebra A with an action of 9x
A -+ A &(9s.
As for usual algebras, corollary 3.3.1 allows us to derive, from such map, a Lie*action
23x Z A
of the Lie*-algebra Z9g
A.(A)
-+
(3.14)
on it. We can now define what it means for the 9-action on
A-mod to be strong.
Definition 3.3.5. Let A be a chiral algebra acted upon by a group Sx-scheme Sx.
This action is called strong if there exists a
9x-equivariant map
,x--A
117
such that the composition
(3.15)
29 M A -+ AM A -A,(A)
coincides with (3.14).
Consider now the action of 9 on the category A-modxi of A-modules on X'.
Let MI be an object in A-mod"S. In particular it is a
SI-module.
Definition 3.3.6. A module MI in A-modjs is called strongly
the two actions of the topological Lie algebra Z I
9-equivariant if
on it coincide. Where the first
action comes from the map
(3.16)
29x -+ A,
and the second action comes from the action of 9 , and by corollary 3.3.1.
Equivalently, we see that MI is strongly equivariant, if the action of 2
from 3.16, can be integrated to an action of the group
A-mod
9
the category of strongly
and by A-mod
9
91.
)
coming
We will denote by
9-equivariant objects in A-mod:
the factorization category given by the assignment I
A-modxi,
-
A-mods.
Consider now the following general set-up. Recall from 3.1.3 that a chiral algebra
morphism
Ad -+ 3
d
defines an equivalence of factorization categories
3-mod(A-mod)
-4
Suppose now that the chiral algebras Ad and
scheme
9
3-mod.
3
are acted on by a group 'Dx-
x in a compatible way, in other words, suppose that the following diagram
118
commutes:
Ae
3 Ae' @ 09, .(3.17)
V
I@Vb®Id
e3C1
0,Cl
0
(')
Suppose that this action is strong. Denote by ki and k 2 the two
9x-equivariant
maps
A",
29x
93d.
and 29x
The commutativity of the above diagram implies that the following diagram also
commutes
idWb Ij
£sx Z
*0
3d'
A(3c)
We have the following proposition:
Proposition 3.3.2. In the conditions above, if the chiral algebra 3 d is in A-modox,
then we have an equivalence
S-mod(A-mod 9 ) ~ 3-mod9 .
Proof. First of all, note that 3d being in A-mod is equivalent to the commutativity
of the following diagram:
Ac
9L.
(3.18)
Z9x
Now, let MI be a strongly equivariant object in 3-mod(A-mod1 ). This is the same
as a collection of objects
MIUKUJ
satisfying the factorization property for 3 with
respect to the finite set K, and the factorization property for A with respect to
the finite set J. However, as explained earlier, being strongly
119
9x-equivariant as a
A-module on X'
is the same as requiring that the Lie algebra action of 2I'
9 x 0 on
the module M, can be integrated to an action of
91. However, when we regard Mr
as an object in 'B-mod1 , and hence we look at the 2(l)
Sx ,O-action on it coming from
the map k2 , the commutativity of (3.18) implies that this action is also integrable.
Moreover, the
9, action coming from it, corresponds to the 91 -action on MI coming
from the weakly equivariance, in virtue of 3.17. This implies that the module M 1 ,
is naturally an object of '3-modfX.
E]
3.3.3
Strong action on the category 'D,-mod
Let g be a simple finite dimensional Lie algebra and with an invariant bilinear form
r.. Recall the chiral algebras Ait and Dit defined in 2.1.3 and 2.4.1. In this section
we will define a strong action of the group Dx-scheme Jx(G) on Ait and Dit. We
will therefore use the notion of "action of a factorization group on a factorization
category" developed in 3.3.
Action of Jx(G) on the chiral algebra A,
Recall the construction of A, given in 2.1.3. It is constructed as the twisted-chiral
envelope of the Lie*-algebra L' = g 9 Dx e
2(X).
We have a natural action of Lg on L' defined by
g - (h + w) = [g,
hLg +
Dx (g, h) + w,
forgE Lg and (h+w) E L,.
Now consider the group-scheme Jx(G). Since the bilinear form
,
is Ad(G)-invariant,
we have a well defined action of Jx(G) on L' given as follows. Let k' be a horizontal
section of Jx(G) over X. This section corresponds to a section k of G over X. Set
120
k' - (h + w) to be
k'- (h + w) := Adk(h) + to(k-ldk, h) +.
Strong action of Jx(G) on 'D,.i-mod
Recall from 2.4.1 the chiral algebra Di.. A better way of describing it is by using
the factorization picture. For this, we can define it in the following way. For every
finite set I, we define 'Deat to be
'Der,:= U(2Z(I)
@0
OJG
,,
U(C4~'d)
where 24), and 2CI'
are the topological Lie algebras over X' attached to L'"t as
defined in (3.8). As it is proven in [AG], 'De, comes equipped with two embeddings
Ac,.t -+ De,.
(3.19)
- Ac, . .
These two embedding endow 'De,. with a structure of chiral Ac,.-bimodule.
In
particular, by considering the right Ac,.-action, we have the following lemma.
Lemma 3.3.1. The chiral algebra D,t is an algebra in Act-mod.
From the factorization description of 'Der, we see that the group Dx-scheme
Jx(G) acts on it via right-multiplication. We also have a natural map
L9 -+ 'Deri
given by the composition
LO 4 Acrit + Dcrit-
It is not hard to see that, from the construction of the right embedding r : Ac,
'Der given in [AG], the Lie*-action of L. on 'D,
121
coming from the above map,
coincides with the Lie*-action coming from the Jx(G)-action on it. Therefore we
have a strong action of Jx(G) on 'Deit, and hence fore, a strong action of Jx(G) on
the category Dit-mod.
Recall that we have defined an action of Jx(G) on Ait whose induced Lie*-L.action is given by the natural map
L9 -4 Acrit
followed by the Lie*-bracket on Ait. Consider now the chiral algebra map
Both Ait and Dit are equipped with a strong action of Jx(G), and moreover, the
maps from L. to Ait and Dit for these actions fit into the commutative diagram
r 'IcritAcrit ----
IZ
LO
Consider now the factorization categories Acit-mod JG and 'Dc,it-mod JG. By lemma
3.3.1 it makes sense to consider the factorization category Deit-mod(Acit-mod
By proposition 3.3.2, we have the following.
Proposition 3.3.3. We have an equivalence of factorization categories
'Deit-mod(Acrit-modJG)
_
Deit-modJG.
In other words, Deit is a strongly Jx(G)-equivariant objects in Acrit-mod.
122
JG)
3.4
The Beilinson-Drinfeld Grassmannian and critically twisted D-modules on it
In this section we are going to introduce the main players of this work: the BeilinsonDrinfeld Grassmannian and the category of critically-twisted D-modules on it. We
will use the language introduced in Section 3.1. In particular, the Beilinson-Drinfeld
Grassmannian GrG will be a factorization space, and the category of D-modules on
it a factorization category.
For a smooth curve X over k, the Beilinson-DrinfeldGrassmannian,denoted by
GrG, generalizes the well-known affine Grassmannian GrG,X classifying G-bundles on
X with a given trivialization outside a point x E X. For every finite set I, we define
a space GrGI over XI. The factorization space GrG is given by the assignment
I
-+
GrGI.
We start by defining, in 3.4.2, the local Beilinson-Drinfeld Grassmannian. We
will then show how the local Beilinson-Drinfeld Grassmannian is equivalent to GrG.
In proposition 3.4.1 we will show how this equivalence allows us to present the spaces
GrGI as quotients of two group schemes over XI. In 3.4.2 we will then define the
category De,.it-mod(GrG) of critically-twisted D-modules on GrG, using the existence
of a canonical line bundle
3.4.1
£criI,I
over GrGI, as explained in 3.4.3.
The Beilinson-Drinfeld Grassmannian
Let G be a semi-simple algebraic group of adjoint type, and let S = Spec(A) be an
affine scheme. Before going into the definition of GrG, we will recall some notions
regarding families of bundles/G-bundles over X. These will be used in 3.21 to obtain
a more manageable description of GrG-
123
For every set I, with
I
=
n, and a map 0 = (#1,...
,#)
SS-P
A X1,
X I,
recall the schemes Dp and D' defined in 3.3 and 3.4.
Definition 3.4.1. For X and S as above, we define the category of gluing data to
be the category of triples (Mx', MD,, 7), where Mx; is a bundle on X , MD" is a
bundle over D, and -y is an isomorphism
Morphisisms in this category are defined as morphisisms of vector bundles compatible with the isomorphisisms 7's.
Consider now a vector bundle M on Xs. The assignment
M -+ (MIxg, MIDo, id)
defines a functor from the category of vector bundles on Xs to the category of gluing
data. Moreover, by Beauville-Laszlo theorem, this functor is an equivalence. In the
case of a Noetherian ring A, this is also a consequence of faithfully flat descent by
looking at the diagram
Dok
D,
>Xs
If instead of considering vector bundles on Xs, we consider G-bundles, the above
statement remains true, and the same functor defines an equivalence
{G-bundles on Xs
}
(PG,X'
Res
-
where y:
124
i PG,D ,7)
PG,X ID'
(3.20)
PG,D, D
J
where PG,X- and PG,D. are G-bundles on X and DO respectively.
We will now define the local and global Beilinson-Drinfeld Grassmannian and show
how the two notions coincide.
Definition 3.4.2. For every finite set I, and test scheme S, we define GrG
S
GrI'(S)
-4
to be
Y
(PG,D4,1)
X,
where y : PG,DI D
,D
D
where PG,D, denotes the trivial bundle on DO.
The assignment
I -+ GrIG
defines a factorization space, called local Beilinson-Drinfeld Grassmannian. The
global version of the above is what is called the Beilinson-Derinfeld Grassmannian
GrG. It is defined in the following way.
Definition 3.4.3. For every finite set I, and test scheme S, we define the space
GrG,I
over X' to be
-4X
Ps
GrG,I(S) =
wher
:PXs
X
PG,Xs X
Clearly, the restriction functor, defines a map
GrG,I
-4
Moreover, if we have a pair (PG,D , y) in GrI,
GrGI-
we can consider the object (P,XC, PG,D, 7
in the category of gluing data. Under the equivalence (3.20), this object corresponds
to a G-bundle on XS, with a trivialization on X , i.e. it corresponds to an object
125
in Gra,1 . In other words, for every I, we have an equivalence
GrG,I ~
GrIOt,
(3.21)
The above equivalence allows us to describe the space Grci as a quotient of a group
ind-scheme by a group scheme. Recall from 3.1.7 the group-scheme JG1 and MG1
over XI. Then we have the following:
Proposition 3.4.1. For every I, the space GrG,I can be described as the quotient
GrG,I ~ MG 1 |JG
1 .
Proof. For every affine scheme S = Spec(A), and S -+ XI, we can regard MG 1 (S)
as
MG1 (S) = Hom(Spec(A((ti,..., tn))), G) ~{ automorphisms of the trivial G-bundle POG D}
where t1 = (t - <* (t)), for t a local coordinate on X.
g E MG1 (S), we can define an element in Gr
1
Therefore, given an element
(S) simply by taking PG,Dg to be
the trivial G-bundle P&,Dg on Dp, and -y to be given by g. This assignment defines
a map
MGr(S) -w>GrIOgrS)
and the fibers are acted simply transitively by the group of automorphisms of
the trivial G-bundle on Dp, which is isomorphic to JG1 (S).
Therefore we have
MG 1 (S)/JG(S) ~ GrI(S), and, by the equivalence (3.21), we have
MG1 |JG 1 ~_+
GrGI.
126
In the next section we will be interested in the category of D-modules on GrG,IHowever, the presentation of GrGi given in proposition 3.4.1 does not make clear
how to define such category. However, as it is shown in [BD2], a remarkable feature
of GrG,I is that it can be represented as the inductive limit of schemes of finite type.
In fact we have the following.
Proposition 3.4.2. The functor GrG,I is represented by an ind-scheme of ind-finite
type (see remark ?? for the definition). Moreover, if G is reductive, then GrG,I is
ind-projective.
3.4.2
Critically twisted D-modules on GrG
We will now define the category of D-modules over the Beilinson-Drinfeld Grassmannian. Since for each I the space GrG,I can be represented by an ind-scheme
of ind-finite type, we start by developing the notion of D-modules on every such
scheme.
Recall that if X is a classical scheme, we have a well defined forgetful
functor 'Dx-mod
-+
QCoh(X). When X is an ind-scheme, we will explain below
the correct replacement for the category QCoh(X) of quasi-coherent sheaves on X
that will be used to construct the ind-version of the above forgetful functor.
We start with the definition of the category QCoh (X) replacing the usual notion
of quasi-coherent sheaves on X. Let X be an ind-scheme X = li
X3. We have a pair of adjoint functors
by ip,,a the closed ambeddings X ,
QCoh(Xa)
Consider the category C :
Xa, and denote
QCoh(XO) : i
.
QCoh(X,,). By definition, we have a map maps
h*
-- QCoh(X,).
Moreover, following [JB], we have the following.
127
Proposition 3.4.3. The functors i : C -+ QCoh(X,,) admit left adjoints in,,,
ia,, : QCoh(Xa) -+ C.
We will denote by QCoh!(X) the category defined as
QCoh'(X) := !L QCoh(Xa).
For two objects T and
maps
#,
: 3
-
P' in QCoh!(X), a morphism
Y' compatible with i
#
(3.22)
: T -+
' is a collection of
As it is shown in [JB], we have the
following.
Proposition 3.4.4. The category QCoh(X) is equivalent to C, i.e.
i
QCoh(X,) ~ m QCoh(X).
Remark 3.4.1. The importance of the above proposition is the following.
The
example of ind-scheme we should have in mind, it that of an affine ind-scheme, i.e
the scheme corresponding to an abelian, complete, separated topological ring whose
toplogy is generated by a ltered system of open ideals {Ii}, s.t. Ii
generated over ii
+ I is finitely
Ij. The ind-scheme X is given as
X = l0
Xi := lSpec(A/Is).
i
i
We are interested in the category of continuous discrete A-modules. This category
is given by the following.
Denote by iij the embeddings i
: X
-
Xi, and consider the functors i,
-+
(A/I)-mod.
functors
4, : (A/Is)-mod
128
as
The category Ac',di-mod of continuous discrete A-modules, is by definition
Acdis-mod:= Im(A/Ii)-mod.
Since we can re-write the above as
i
QCoh(Xi), it seems natural to think that
this is the right category to consider. However, the presentation of it as a limit,
has the disadvantage that it is not clear how to compute maps out of it. However,
proposition 3.4.4 says that the category Ac,dis-mod can also be described as the
colimit under the maps
44,j,, : (A/Ij)-mod -+ (A/I;)-mod.
We will now define the category of D-modules on an ind-scheme X = 1iXa,
with X' s schemes of finite type. Recall that for a closed embedding is,3
:X
-+ Xp,
we have a natural exact functor
: Dmod(X,)
satisfying Fp o i,,
D-mod(X,)
-+
=
o0,,,F,
-+
D-mod(Xp),
where Fy denotes the forgetful functor F,
QCoh(Xy). We define the category D-mod(X) to be
D-mod(X) := I'
D-mod(Xa).
Note that we clearly have a forgetful functor
F: D-mod(X) 129
QCoh!(X).
Consider the Beilinson-Drinfeld Grassmannian GrG given by the assignment
I -+ GrG,I.
By proposition 3.4.1 and 3.4.2, we have:
" The functor GrGI can be represented by the quotient MGI/JGI.
" The functor GrGI is represented by an ind-scheme of ind-finite type.
We will write GrG,I as
GrGI
1YI
-
In virtue of the above, it makes sense to consider the category QCoh!(GrG) given by
I
-+
QCoh!(GrG,I),
and the category of D-modules on GrG defined as
D-mod(GrGi) := li
The assignment I
-+
D-mod(Y').
(3.23)
D-mod(GrG,I) defines a factorization category, denoted by
D-mod(GrG).
Twisted D-modules on GrG
We will be interested in the category of twisted-D-modules on GrG. These are defined as modules on GrG endowed with an action of a sheaf of twisted-differential operators on GrG. In particular, we will be interested in the category Dc,--mod(GrG)
of critically-twisted differential operators on GrG. We will now recall the definition
of these objects.
130
Recall that, as it is explained in [BB], given a Picard algebroid on X, i.e. a sheaf
of Lie algebras ' on X, such that
0
-+
O x -+'P -+ Tx -+ 0,
and such that, for any r and r' in '
and
f
E Ox we have
[+,
frf]
=
f[r
rf]
±
we can consider the algebra D,. This is the universal algebra equipped
with morphisms i :x
.
D, and i :'P
-"
Dj, such that
" i is a morphism of algebras.
" i, is a morphism of Lie algebras.
" for f E Ox,r E 'P one has ij,(fr/) = i(f)iT(q) and [i,(7),
i(f)i
=
(7)f
We call D, a sheaf of twisted differential operators on X.
Consider now a line bundle L on X, and the algebroid 'P defined as the algebroid
of G1Im-invariant vector fields on the principal Gm-bundle associated to Z. We have
maps
0 -+ Ox -+ 'PL -+ Tx
-+
0,
making 'P a Picard algebroid over X.
Definition 3.4.4. We define the category D,-mod(X) of L-twisted Dx-modules to
be the category of Ox-modules endowed with an action of the sheaf Dj .
Note that we have an equivalence of categories
'Dx-mod ce Dz-mod(X)
given by M -+ M @ Z.
131
Critically twisted differential operators on Gra,I
Let's now go back to the space Gro,I = MG 1 /JG
D-mod(GrG,I) as defined in (3.23).
over X'
and the category
For every I, we will construct a line bundle
£crit,I
£erit,I -+
GrG,I,
and consider the category Derit-mod(GrG,I) defined as
Dcit-mod(GrG,I) := Dzcr-mod(GrG,I)-
(3.24)
As we pointed out before, we have the following proposition
Proposition 3.4.5. For every I, there exists an equivalence of categories
D-mod( GrG,I) ~-Deit -mod( Gra,1),
given by M 1
3.4.3
-+
M1 0
"erit,1.
Construction of the line bunde Z i over the BeilinsonDrinfeld Grassmannian
We will first recall the definition of the line bundle Lcit,x over the affine Grassmannian GrG,, presented in [BD2].
We will then generalize this construction for the
spaces GrG,I.
We will start by recalling some definitions from Tate linear algebra.
Definition 3.4.5. A Tate vector space V is a complete topological vector space
having a base of neighborhoods of 0 consisting of commensurable vector subspaces.
* A subspace P c V is bounded if for every open subspace U C V there exist a
finite set {vi, ... ,Vn} E V such that P
132
c U + kv 1 + - - - +
kVn.
*
A c-lattice in V is an open bounded subspace.
* A d-lattice in V is a discrete subspace F C V, such that V = F + P, for some
c-lattice P C V.
Let x be a point in X and t a coordinate around it. Consider the formal disc
DX, and the formal punctured disc Do. Denote by Q1 (9) the ring C[[t]] and by X
the field C((t)).
Example 3.4.1. Given a vector bundle Q on X equipped with a non-degenerate
symmetric form
QQQ
--+
Qx,
and a point x E X, we can consider Q@Q1 (9)
C QX. The vector space V := QOX
is a Tate vector space, moreover it is equipped with a symmetric nondegenerate form
given by the residue. The subspace L := Q0 Q(Z) is a c-lattice in it. Moreover it
is a Lagrangian subspace.
More generally, for every non-empty finite set of points S C X, we have the Tate
vector spaces and corresponding Lagrangians,
L := DxceSQ @
Q1)x
C V := DxesQ 0 X.
As a special case of the above example, given a square root Z of the line bundle
QDx,
and a vector space W with a non-degenrerae symmetric form
W x W -+ C,
we can consider the vector bundle Q := Z 9 W. Let now W be equal to the Lie
133
algebra g, and consider the killing form n'Kiu on g. The Tate vector space Vz,
Vz :=L®@r (g@X),
carries a non-degenerate bilinear form given by
VZ (9 V"'> -L
Z
(9 Z (9
~-
QD
-
Consider the Lagrangian subspace Lz, equals to
Lz :=Z 0g.
Denote by Cl(V) the Clifford algebra associated to V and by M the irreducible
Cl(V)-module
M := Cl(V)/Cl(V)L.
Denote by Lagr(V) the ind-scheme of Lagrangian c-lattices in V = Vz as defined in
[BD2] 4.3.2.
There exist a canonical line bundle 'PM on Lagr(V) defined as follows.
Definition 3.4.6. We define 'M to be the line bundle over Lagr(V) whose fiber
over L' E Lagr(V) is
PM,L' = ML' =
{m E MI L'- m = 0}.
Consider now the map #: G((t))/G[[t]] -+ Lagr(V) given by
g
-+
gLg- 1 .
Following [BD2] 4.6.11 we have the following definition.
Definition 3.4.7. We define the line bundle Zit,x on GrG,z to be the pull-back,
134
along
#
of the line bundle 'PM;
,-uri,2
:=
4*'PM
- GrG,x -
We will now try to generalize the above to powers of X.
In particular, we will
construct a sheaf of Tate vector spaces over X'.
Let us fix a square root Z of the canonical bundle Qx. Consider the Dx-module
Vx given as
Vx = L Ox (g O'Dx).
As before, the killing form on X together with the fact that L 0 L ~ Qx, defines a
symmetric bracket
(3.25)
Vx 9 Vx -4 A, (Qx).
In particular, we have a skew-symmetric pairing on Vx[1], and therefore a Lie*algebra structure on the direct sum V[1] e Qx. Define
Cl(Vx) to be the twisted
enveloping chiral algebra of V[1] o Qx,
Cl(Vx) := U'(V[1] (Dfx),
where we regard V[1] as a commutative Lie*-algebra. For every finite set I, consider
the sheaves of topological vectors spaces
L(I) = hr (p(Vx)[-I])
and
V(I) := hr(jj*(P!(Vx)[-I])),
where the functor hr is the one defined in 3.8. The map (3.25) gives us a map
(, )(I) :
0)
&
-+
hr(jej*(P!(Qx)[-I]))-
Note that, if we take the fiber of V(V) and L(M) at (Xi,
135
...
,xn)
(3.26)
E X', we recover the
Tate vector space and the c-lattice from example 3.4.1. Moreover the map (3.26)
becomes
( Xi) x (E1Q 0 X)
(Resi
ResC
x
x C.
(3.27)
The above topological vector spaces will be our family of Tate vector spaces.
More generally, we have the following definition.
Definition 3.4.8. Let R be a commutative ring. A Tate R-module is a topological
R-module isomorphic to P D
Q*,
where P and
Q
are infinite direct sums of finitely
generated projective R-modules.
" A c-lattice in a Tate R-module V is an open bounded submodule P C V such
that V/P is projective.
" A d-lattice in V is a submodule F C V, such that for some c-lattice P, one
has Frn P = 0 and V/(F
+ P) is a projective module of finite type.
Let's go back to the topological vector space V(') over X(').
Definition 3.4.9. We say that a c-lattice L' c V(') is Lagrangian, if for any geomet-
ric point (X1 , .
.
, Xz)
E X', we have that the maps in (3.27) define n non-degenerate
forms on the quotient V(I)/L'.
Denote by Lagr(V(')) the ind-scheme of Lagrangian sub-spaces in V(I). In particular, note that L(I) E Lagr(V(')). Consider now the chiral Cl(Vx)-module over
X' equal to
M1 = Cl(Vx) 1 .
From the definition of
el(Vx),
proposition 3.1.4, and the fact that we have maps
VI) -+ hr(jij*(p1(Vx @ Qx)[-])),
136
we see that V(I) acts on any chiral-Cl(Vx)-module on XI. In particular it acts on
MI. Similarly to the above, we have the following definition.
Definition 3.4.10. We define the line bundle 'P,
on Lagr(V(')) to be the line
bundle whose fiber over L' E Lagr(V(I)) is
'PI,,y = ML' = {m E Mil L'- m = 01.
The adjoint action of G on g, defines an action of MG, on V('), that we will still
denote by Ad. Consider now the map
#1:
MG 1 /JG
-+
Lagr(V(')) given by
g -+ Adg(L(I)).
Definition 3.4.11. Define the line bundle Leritj over GrG,I ~ MG 1 /JG
thepull-back along
#,
to be
of 'MP;
Scrit,I :=
*Pmj -+
GrG,I-
We therefore arrive to the following definition.
Definition 3.4.12. Let GrG be the Beilinson-Drinfeld Grassmannian. We define
the factorization category De,.it-mod(GrG) of critically twisted D-modules on GrG
to be the category given by the assignment
I -4 Dcrit-mod(GrG,I) = Dccrt -mod (GrG,),
where Dz,,-mod(GrG,I) is defined as in definition 3.4.4.
The reason why they are called critically twisted is given by the following propo-
sition (see [BD2]). Let - be the Lie algebra given in 2.1.1.
137
Proposition 3.4.6. Denote by ir the projection
,r : G((t)) -4 GrG,X.
Consider the pull back along 7r of the line bundle Leit,x. Denote by G((t)) the corresponding Gm-bundle on G((t)). Then, the Lie algebra correspondingto the extension
1 -+ Gm, -+ G((t)) - G((t)) - 1,
is equal to
0- 4
crit
-4 g((t)) -+ 0,
where 'cit denotes the Kac-Moody algebra at the critical level
3.4.4
D-modules on the Beilinson-Drinfeld
Ierit =
-1/2ki.
Grassmannian
as chiral 'D,.-modules
Consider the factorization category 'Dcrit-mod JG
=
{I -+ Dit-modJ}
of strongly
JG-equivariant 'Dc,it-modules. We want to relate this category to the factorization
category Deit-mod(GrG)
=
{I -+ Dcrit-mod(GrG,I)} of critically-twisted D-modules
on the Beilinson-Drinfeld Grassmannian GrG- We start by considering 'Derit-modules
supported at some point x E X, where this relation is completely understood (see
[AG]). We will then pass to the categories 'Deit-modiG and Dc,it-mod(GrG,I) over
XI.
The equivalence over the point
Recall from [AG], that to specify a structure of a chiral 'Deit-module supported at
x on a vector space M is the same as to endow it with continuous (w.r. to the
discrete topology on M) actions of Q'('R)G((t)) and -cit compatible in the sense that
138
for 71 E -cit,
f
E OG((t)) and m
e
M,
. (f.m) = f.(77.m) + Lie7 (f).m,
where 77 is the corresponding left-invariant vector field on G((t)). This follows from
the construction of 'Derit and from the fact that M, when viewed as a chiral module
-ass,x
for Jx(G) supported at x, becomes a module for Jx(G)
_-
,
and that
ass,x
(3.28)
!G((t))~
Jx(G)
The right embedding of Ait into 'Dit given by (2.18), endows M with a structure
of right
This action is compatible with the OG((t))-action, in the sense
cit-module.
f
that for ( E -it,
E OG((t)) and m E M,
(.(f.m) = f.(.m) + Liegr(f).m,
where (' is the corresponding right-invariant vector eld on G((t)).
Consider now the category Dit-modx defined as
D,it-mod JG)
'De-it-modG[ [t]:
In other words, we are looking at those 'D,.it-modules at x on which the right action
of g[[t]]
c
cit
can be integrated to an action of Jx(G)2
=
G[[t]]. In the above, we
regard a module M E 'Deit-modx as a g[[t]l-module by means of the right action of
on it and the fact that the sequence
0+0-+
,.
- g((t)) -+ 0,
splits over g[[t]].
Consider now the affine Grassmannian GrG,x = G((t))/G[[t]], and the category
139
Dc,-it-mod(GrG,x) of critically twisted D-modules on it. Recall that, by subsection
3.4.2, this category is isomorphic to the colimit
Dc,it-mod(GrG,x,) := I' D-mod(Yi),
where GrG,X
=
QCoh(GrG,2).
I Yi. In particular, recall that we have a forgetful functor D,.it-mod(GrG,z)
We can describe the category D,.it-modu[ll as D-modules on GrG,z-
In fact, we have the following proposition.
Proposition 3.4.7. There exist an equivalence of categories
Derit-mod( GrG,x)
-Derit
-mod G[ [t||
Proof. The proof of the above proposition can be found in [AG]. However it is useful
to recall how the functor is constructed. Let M be an object in Dcit-mod(GrG,x),
and denote by r the projection r
: G((t)) -+ GrG,x. Consider the pull back 7r*(M).
We can define on the vector space P(G((t)), 7r* (M)) a structure of chiral 'D,.it-module
at x in the following way. The module F(G((t)), 7r*(M)) is naturally a discrete (9 G((t))module, and therefore, by (3.28) a Jx(G)-module supported at x. Moreover, the
projection 7r is right-G-invariant, the right D-module structure on M, gives rise to
the action of g((t)) on 7r*(M), therefore, F(G((t)),7r*(M)) is indeed a chiral Ditmodule supported at x.
The fact that it belongs to D,it-mod ft]l follows from
noticing that the right action of g[[t]] on it coincides with the G[[t]]-action coming
from the G[[t]]-equivariant structure on 7r*(M).
E
The equivalence over X,
Let's now consider the category 'Deit-mod, of 'D,it-modules on XI. Consider the
Jx(G)-action on Dit as defined earlier. Recall that the Lie*-L,-action coming from
140
-+
the Jx(G)-action coincides with the Lg-action coming from the composition
L94 Ac.t r 'Dcrit.
We are interested in the category Dit-modJx(G) of strongly Jx(G)-equivariant
objects in 'Dit-modi. Objects in this category can be described as modules M1 E
Dit-mod, on which the Lie action of
24J0
can be integrated to an action of the
group scheme JG 1 over XI. Note that, by considering the case of I
the discussion before, where 20,
=
0, we recover
is exactly g[[t]].
We will start by describing the category 'De,.it-mod, in a more suitable way. Recall
the group ind-scheme MG, of meromorphic jets defined in (3.6). Let p' be the map
p' : MG 1 -+ XI,
and denote by P MG1
rel the functor
o
pMGI
QCoh (MGI)
-e>
{
discrete 9MG 1-modules}
Note that this functor corresponds to the functor F(G((t)), ) if we take the fiber at
E X for 1= {*}. Denote by Ore,
the sheaf of topological algebras over X' given
as
PMG'(OMG1 )
0
MG-
We have the following proposition.
Proposition 3.4.8. To specify a structure of a chiral 'De,.it-module on X,
on a
quasi-coherent sheaf M 1 is the same as to endow it with continuous (w.r. to the
discrete topology on M) actions of 0"G and r
141
compatible in the sense that for
7 E 2'I',
f
E OJG1 and m G MI,
'7.(f.m) = f.(7.m) + Lie 7 (f).m,
where nq is the corresponding left-invariantvector field on JG 1 . Where 23a denotes
the sheaf of topological Lie algebras defined in (3.8) for L = Lot .
Proof. From the construction of 'De,.,
we see that a chiral 'De,.-module M
on
XI, is in particular a module for U((9Jx(G))/K, where K is the kernel of the map
U(OJx(G))
-
OJx(G).
Since, for a Lie*-algebra L, chiral modules for U(L) on X,
are in bijection with Lie modules for 2' defined in (3.8), we see that MI is naturally
a Lie*-module for the commutative topological sheaf of Lie algebras over X,
hr(jj*(pi(Jx(G))
[- 1(3.29)
However the sheaf of topological algebras in (3.29) coincides with 0(G1, therefore
MI becomes a continuous (9rd -module, i.e. a module over the group ind-scheme
MG.
Moreover this is an equivalence between chiral modules for U((fJx(G))/K
over X- and discrete (g9l
-modules, i.c. objects in QCoh(MGI), as explained in
corollary 3.1.1. Now, from the definition of 'D,
in factorization terms, it is also
clear that MI comes equipped with a 2(4) -action and that the latter needs to be
compatible with the former (9MGc-action.
[]
Consider now the Beilinson-Drinfeld Grassmannian GrG. As it is explained by
proposition 3.4.1, for every finite set I, we can describe the space GrG,I as
GrG,I c- MG 1 /JG1 .
142
The above quotient is an ind-scheme,
GrG,I = I'
i 7
and, as before, each Y,' is of finite type.
Consider now the category Dc,it-mod(GrG,I) of critically-twisted D-modules on
it. Denote by 7r, the projection
iri : MG,
--
GrG,I.
For a D-module Th in D,it-mod(GrG,I), consider the pull-back
ir(7 1 ).
This is, by
definition, an object in QCoh(MG1 ), and, therefore, the object
FMGI (7;r
Tel
is a discrete module for 0G
I
(,T))
(see example 3.1.3 for the definition of (JG). We
claim that there is a natural 'De,.it-module structure on MI. In fact, we have the
following theorem.
Theorem 3.1. There exist an equivalence of factorization categories
D,it-mod(GrG)
' Deit-mod!"
given by T1 _ rMGI(71(T)).
Proof. As we explained before, the object MI = ]pMGI (7j (TI)) isa discrete
Gr-
module, and therefore, a chiral U(Jx(G))/K-module over XI. Now, the (negative)
of the action of
..
on T 1 gives rise to an action of the same Lie algebra on MI,
compatible with the OrG,-action. Therefore, by proposition 3.4.8, the objects M,
is a 'Deit-module on XI. Now we claim that the right action of Z,it on MI coming
from the right embedding of Acit, is obtained by derivating the JG 1 -action on -r;(71)
143
coming from the equivariant map -7r. This would imply that M, is indeed strongly
Jx (G)-equivariant. This fact is proved by repeating the argument presented in [AG]
Proposition 6.7.
144
3.5
The space of Opers
In this section we will recall the definition of Opers as given in [BD2]. In particular,
given a curve X and a point x E X, we will recall the definition of opers on the disc
D, (resp. punctured disc D') and its presentation as a scheme (resp. ind-scheme).
We will also generalize the above definitions in order to obtain factorization spaces
and study the factorization categories of modules over them. In 3.5.1 we introduce
the factorization space Op. corresponding to the 'Dx-scheme of opers. In 3.5.2 we
construct the factorization space Op* corresponding to opers on the puncured disc,
and show how this can be represented as an ind-scheme. We will then introduce the
factorization space Op"' of unramified opers.
3.5.1
The space of opers
Let X be any smooth curve, G a simple algebraic group of adjoint type, and B C G
a Borel subgroup. For a B-bundle PB on X, denote by PG the induced G-torsor
PG = G x PB. We have the corresponding twisted Lie-algebras bp := bp, = b x PB
B
B
and gG := 9PG = g X PG - g x PB.
The Lie algebra gG is equipped with a
B
G
standard filtration, induced from the filtration on g given by the choice of b. We
have g-r = g, and gi+1
=
[9 i, n]; in particular we have g0 = b and g1 = b. Let now
V be a connection on PG. For any connection V' preserving PB, we can think of
V - V' as an element in 9G 0 QX
= 9B
0
X
We denote by c(V) the projection
onto (g/b)B 0 Qx
c(V) := (V - V')
mod bB.
Definition 3.5.1. An oper on X, is a pair (PB, V), where PB is a B-bundle on X,
V is a connection on the induced G-bundle PG satisfying the following:
. c(V) E (0- 1 /b)B
0 Qx C (g/b)B 0 Qx.
145
* For any simple negative root a, we have that the a-component c(V)a E
OG
*@
Qx does not vanish on X.
Equivalently, we can think of opers in the following way. Lets choose a trivialization of PB, and let Vo be the tautological connection on it. Denote by H the
set of simple roots of g. Then, as it is explained in [FG3], an oper is given by an
equivalence B(X)-class of connections V of the form
V = Vo +
#E fe, + q,
aErI
where each
#,
is a nowhere vanishing one-form on X, and q is a b-valued one form.
Changing the trivialization of PB by g : X -+ B, the connection V get transformed
into V' = g-'Vg - g-'dg.
The above makes sense in families. Indeed, if S is a Dx-scheme S
4
X, then
we have a well defined notion of G-bundle with a connection V along X.
G-bundle PG
i.e.
It is a
4 S on S, such that PG is a Dx-scheme and the map 7r is horizontal,
a map of 'Dx-schemes. We define opers over S to be the set consisting of
pairs (PB, V), where
PB
is a B-bundle on S, and V is a connection along X on the
induced G-bundle PG such that the conditions above are satisfied, with Qx, replaced
by
#*(Qx).
It can be shown that the above functor is represented by an affine Dx-
scheme, denoted by Op,,X. According to 3.1.6, we therefore obtain a factorization
space {I -+
O
pg, := JOp,,X} that we will simply denoted by Opg. Therefore we
have spaces Op,,, over XI, where for any test scheme S,
S
Opg,(S) =
is
X', (Pa, PB, V),
where PG is a G-bundle on Dp,
PB
is a reduction to B,
and V is a connection on PG, satisfying the oper condition
Note that, if we take S = Spec(k), and I to be the set with one element, then
Opg,x(S) =: Op,(Dx) is the space of regular opers introduced in [FG3].
146
By con-
struction we have the following.
Proposition 3.5.1. The assignment I -+ Op,,
space Op,. Moreover,
0
defines a co-unital factorization
p,,X is affine, in particularit correspond to a commutative
chiral algebra on X.
3.5.2
Opers on the punctured disc
Recall now example 3.1.3. In particular recall that, for an affine 'Dx-scheme
have defined spaces M 1
we
over XI, that contain J 1 9, where
S
MA (S)
,
''''>X
,
:=
a horizontal section D
-+
Z
where Do is the scheme defined in 3.4. This construction, in the special case of
=
Op,,x, generalizes the notion of opers on the punctured disc Op.(D*) introduced
in [BD2]. It is defined in the following way.
Definition 3.5.2. For every I finite set, and test scheme S, we define OpI, to be
the space over X' given by
S is
Op",*(S)
=
'"
XI, (PG, PBV),
where PG is a G-bundle on D', PB is a reduction to B
and V is a connection on PG, satisfying the oper condition
Note that, if we take S = Spec(k), and I to be the set with one element, then
Op,,X(S) = Op,(D.*). In particular, from example 3.1.3 we have the following.
Proposition 3.5.2. The assignment I -+ Op.,
defines a factorization space Op,.
Remark 3.5.1. Note that, as expected, the above factorization space, in contrast
to Op., is not co-unital. However, as it is explained in corollary 3.1.1, if we consider
147
the chiral algebra
op,,, then we have an equivalence
(0p.,,
-oduls
Chirl
O~el
modules for the topological
Xcontinuous
~associative
"'
where
o
Oxi-algebra 0"
was defined in example 3.1.3.
Relation with the center
3rit
Recall the commutative chiral algebra
3
cit
defined as the center of Acit. Recall that
this chiral algebra is related to the space of G-opers, where G denotes the Langlands
dual group of G. In fact in [FF] they prove the following theorem.
Theorem 3.2. For the critical level ,.it, we have an isomorphism of chiral algebras
3
crit ~
(0sx)-
The above theorem allows us to describe the category of chiral 3,.it-modules
over X' in terms of modules over the topological algebra Ord.
From the previous
remark we have the following equivalence
3cit-modi
{
Chiral 3,t-modules on XJ
4
f
continuous modules for the topological
associative (xi-algebra 0 e19
(3.30)
We will denote by QCoh'(Op,1) the category on the right hand side of the above
equivalence. We will denote by QCoh!(Op') the factorization category given by the
assignment
I -+ QCoh!(Opi,).
Equivalence (3.30) can be regarded as an equivalence of factorization categories
3cit-mod ce QCoh!(Op').
148
3.5.3
Unramified opers
Recall now the sub-functor Op"' c Opg(D ) of unramified opers on D2 introduced
in [FG5] and [FG2]. It consists of opers on D*, that are unramified when regarded
as G-local systems. In other words, are those pairs (PB, V) such that V is G((t))Gauge equivalent to the trivial connection Vo = dt. As it is explained in [FG4]
the space Op,n can be described as a closed sub-scheme of Op9 (Do), in particular,
the algebra of functions
9
( opunr
on Op"' has a structure of a topological algebra.
t
We denote by QCoh(Opu"
) the category of continuous discrete modules over this
algebra. We will now define the factorization space corresponding to OpuTn.
Definition 3.5.3. For every I finite set, and test scheme S, such that S
X1, we define the space Op,
Op,g,\ (S) =
over X' by
(PB, V), where (PB, V) is an oper on D',
0
and the pair (PG, V) can be extended
to the entire D,
Note that, if we take S = Spec(k), and I to be the set with one element, then
Opu,"(S) = Opu,".
From the definition, we have the following lemma.
Lemma 3.5.1. The assignment I-+
Opu?""
defines a factorization space Op
Remark 3.5.2. It can be showed that the algebra (
,1
nr
has a structure of a
topological algebra over XI. As before, we denote by QCoh(Op,r) the category
QCoh! (Opur)
{continuous
J
modules for the topological
associative Oxi-algebra
3.6
(9U unr
The Conjecture
Recall the Drinfeld-Sokolov reduction Tx as defined in 2.3.1. Consider the category
Acit-mod JG consisting of strongly JG-equivariant chiral modules on X as defined
149
in 3.3.6. Consider the restriction of Tx to this category
Tx: Ar.,it-mod JG -+ 3c,.it-modx.
The above functor has been studied by D. Gaitsgoy and E. Frenke. In [FG2] they
show the following.
Theorem 3.3. Let Tx be the Drinfeld-Sokolov reduction at the critical level.
1. The
functor Tx,
restricted to Ac,.it-modcG is exact.
2. It defines an equivalence of categories
Tx: (A,it-modx)JG 4 QCoh( Op,
)
where we regard QCoh!(Op4) as a sub-category of 3c,.it-modx via the equivalence 3.30 and the inclusion OpnX C Op,
3.6.1
The conjecture over X,
Consider the factorization category A,.it-mod = {I
-+
Ac,.it-modi}. Recall that in
3.3.3 we have defined a strong action of the group 'Dx-scheme Jx(G) on the category
A,.it-mod. For every finite set I, we are interested in the category A,-it-mod/G of
strongly Jx(G)-equivariant objects in A,.it-mod1 . Conjecture 1.2.3 states that we
have a description of this category similar to the one provided by theorem 3.3.
In order to state the conjecture, in the next sub-section we define the Drifeld-Sokolov
reduction T, over X, as a functor
1F: Acit-modj -+ 3,it-mod1.
150
Using 3.30, the functor 'IP can be seen as a functor
1 : Ait-mod, -+ D(QCoh!(OpOI)).
The assignment I -+ Wl defines a factorization functor IF
T : A,.it-mod -+ D(QCoh'(Op')).
We consider the restriction of IQ, to Ac,it-mod 1,
Tr : Acit-modG
-+
D(QCoh (Op?,I)).
The main conjecture is that the same equivalence as the one in 3.3 holds for modules
over X 1 .
Conjecture 3.6.1. Consider the functors I,
T1 : Ait-modi G
1. The above
-4
D(QCoh (Opi)).
functor is exact.
2. The image of Ti is contained in QCoh(Opu" ).
3. The collection of functors T = {I -+
'I}
establishes an equivalence of factor-
ization categories
Ac,it-modJG 4 QCoh ( Op"'').
151
Drinfeld-Sokolov reduction for modules over X'
Recall from section 2.3.1 the BRST-reduction. Recall that, for any Lie*-algebra !
and any map of chiral algebras
f
: U(Z-Tate) -+ 'R, we defined a functor BRST,
BRST: R-modx -+ BRST('Z ® el(2))-modx.
We will start by generalizing the BRST reduction to modules over X'. In fact, the
construction of I' follows from the possibility of extending the BRST-reduction for
A = U(2-Tte) 0 el'(2)-modules over X to A-modules over XI. All we have to do,
is to be able to define a differential on any A-module M, over X,
BRST-reduction for modules on X'
Lets M, be a chiral A-module on X' and let us regard it as a Lie*-A-module. By
proposition 3.1.4, M1 is therefore a module for the topological Lie algebra A,)0
-
hr(pi(A)[-I]) (see Definition 3.8). We define the differential dM, on M, to be the
action on it of the element
Q(),
where
Q(I) E A()
is the section on A(,) corresponding to the image of the identity endomorphism
under the map X :
0 1* -+ A 1 [1], where x is the map defined in 2.14. The pair
(MI, dm,) is naturally a BRST(A,)-module on X'. If we are given a map of chiral
algebras
f: U(2j-Tate)'
-4
'R and a Az-module M, on XI, then its BRST-reduction
will be a BRST(AT)-module on X'. Therefore, for every I, we have functors
{Aqz-modules
on X'} BRST1> {BRST(A3)-modules on XI}.
We have the following proposition.
152
Proposition 3.6.1. Given a map of chiral algebras f : U(2-Tate)' -+' 9, the assignment I -> BRST defines a factorization functor
BRST: Am-mod
-+
BRST(Az)-mod.
Proof. For every surjection < : I -* J, and for every partition 7r : I -w J of I, we
need to show that
Ap o BRST~ BRSTj o A*,
BRST|U(i/)
~ (BRST, 0
...
0 BRST)Iu>iJ),
(3.31)
where I = LgIj. These both follow from the fact that the topological Lie algebra A,
factorizes, and we have A*-(Q(,))
Q(J), and
of the corresponding Q(I')'s in A
Q( Iup>j
corresponds to the product
.
If we consider the natural map U
a)'
A,it, and the chiral Cl(22)
el(2.) 1
over XI, then, given an Ac,.it-module M 1 , we can consider the corresponding Act 0
Cl(.22)-module M 1 0l(22)
1.
Therefore we have functors
BRST : Ait-mod1
-+
BRST(Ac,.it eCl(2))-mod1 .
As we have explained in 2.3.1, we can furthermore modify the differential using the
character x to obtain the Drinfeld-Sokolov reduction xI1,
I1 : A,it-mod,
-+
BRSTx(A,it 0 Cl(.2))-mod 1 ~ 3erit-mod 1 ,
where the last isomorphism follows from theorem 2.2. Now, according to (3.30), we
can rewrite the above as
I,:
Acrit-mod, -+ D(QCoh!(Op?,1,)).
153
3.7
The localization Conjecture for the BeilinsonDrinfeld Grassmannian
In this section we define a factorization algebra B and a factorization functor FL :
Deit-mod
-+
B-mod(QCohl(Op?)). Assuming conjecture 3.6.1, we show that [>,
induces an equivalence of categories between D-modules on the Beilinson-Drinfeld
Grassmannian, and the factorization category of B-modules in QCoh'(Opr"'), as
stated in conjecture 1.2.2.
Recall the factorization category Acrit-mod, the factorization space Op' and its
factorization sub-space Opn". Consider the factorization categories QCoh'(Op?)
and QCoh!(Op'n) as defined in 3.5.2 and 3.5.3. Recall the functor TII,
'Tr Acrit-mod, -+ D(QCoh (Op?,j)).
By conjecture 3.6.1, the above functor, restricted to the category Acrit-mod!G is
exact and induces an equivalence of categories
Acit-modiG
3.7.1
-
WI
QCohl(Op"/).
Definition of the functor IT
Recall the chiral algebra 'Derit from 3.3.3.
By Lemma 3.3.1, the corresponding
factorization algebra is a factorization algebra in Acit-mod, and moreover it has a
natural action of the factorization group JG, making it an algebra in Acit-modJG
We can therefore consider the factorization category
Dcrit-mod(Acrit-mod JG)
154
where, for each finite set I, we take Dcit-mod(Ait-mod JG) to be 'Deit-mod(Ac,it-modiG)
as defined in definition 3.1.8.
Since 'Dit, is an object in Acit-mod J
it makes sense to consider the object 31,
where
'31 := 1('Derit,1).
By definition, the assignment I -+ '3 defines a factorization algebra in QCoh!(Op').
However, by the second point of conjecture 3.6.1 it is in fact a factorization algebra
in QCoh
(Op,nr).
We will consider the factorization category
'3-mod(QCohl (Op,nr))
as defined in 3.1.8. For every I, the composition
D,.it-mod(Acit-modiG)
Ait-modiG
4 QCohl(Opb"),
lifts to a functor, that we will still denote by TI,
TI: Dit-mod(Arit-modJG) -+ 3-mod(QCohl(Op "/)).
We denote simply by IFthe collection of functors {I
-+
T};
F: 'Deit-mod(Acit-modJG) -+ 3mod(QCoh'(Op,"r)).
(3.32)
Recall now the Beilinson-Drinfeld Grassmaniann GrG, and the factorization category
Dcrit-mod(GrG) of critically twisted D-modules on GrG, given by the assignment
I
-+
Deit-mod(GrG,I).
In proposition 3.3.3 and in theorem 3.1 we have shown the following two facts.
155
*
We have an equivalence 'De,it-mod(Acrit-mod JG)
Dcrit-mod
JG
* There exist an equivalence of factorization categories
Dcit-modJ
Deit-mod(GrG)
Recall that the second equivalence is constructed as follows. For every I, denote by
7r, the projection
: MG,
Ari -+ GrG,IFor a D-module Y1 in Dc,it-mod(GrG,I), consider the pull-back ir(Y,). This is, by
definition, an object in QCoh(MG1 ), and, therefore, the object
M
i-
ae
dicrt
=
(7tg (,T)) :=
-,MGI (7X (.T))
(modul)
for OrelIh
G = pI(OMG1 ), where p, is the map p:
is a discrete module for
MG1 -+ XI.
The equivalence above is given by the functor F
I, : Deit-mod(GrG,I)
:7-
cit-moG
-
a
r
GI (7XJ (,T,)).
Let's now consider the composition
Deit-mod(GrG,I)
r,
Drit-mod!G
c4
3crit-mod(Acrit-mo
G)
-+'-mod(QCohI(Op,,r))-
Denote by Fr,, the resulting factorization functor
,: it-mod(GrG,I) ~+'3-mod(QCoh(Op ")).
156
We can finally state the main corollary of conjecture 3.6.1, from which conjecture
1.2.2 will follows using the equivalence
D-mod(Gra,I) ~ Deit-mod(GrG,I),
of proposition 3.4.5.
Conjecture 3.7.1. The collection {I -+ Fq, 1 } gives rise to an equivalence of factorization categories
Deit-mod(GrG) ~> 93-mod( QCoh'(Ope"L)).
(3.33)
r.
We will prove theorem 3.7.1 assuming the conjecture 3.6.1
proof of conjecture 3.7.1. By Theorem 3.6.1 2, the chiral algebra 3 is an algebra in
the factorization category QCoh'(Opf"). In particular, for every I, it makes sense
to consider the category 3-mod(QCohl (Op,)).
Recall now that the functor 1P,7
is constructed as the composition
Dcrit-mod(GrG,I) 4
,it-mod JG
,it-mod(Ac,-it-mod iG)
4 3-mod(QCoh'(Opu"/)).
However, conjecture 3.6.1 3. implies that the last functor gives rise to an equivalence
Derit-mod(Acit-modJG) 24 3-mod(QCohl(Op ,"/)),
therefore, we indeed get an equivalence
Dit-mod(GrG,I)
4
93-mod(QCoh1(Op,r)),
over XI, for every I, i.e. we have that Fy establishes an equivalence of factorization
categories.
157
El
As we mentioned before, using the equivalence between Deit-mod(GrG) and
D-mod(GrG) we arrive at the algebraic description of the category of D-modules on
the Grassmannian GrG and of the category of D-modules on the affine Grassmannian
GrG,.-
Theorem 3.4. The composition
D-mod( GrG) * "'> Deit-mod(GrG)
2-mod(QCohg(QOpgn"))
is an equivalence of factorization categories.
Corollary 3.7.1. The functor M -
,
Lcit,. ) establishes an equivalence of
(
categories
D-mod( GrG,x) where
3
3
-modun,,x
-modun, denotes the category of 3-modules supported at x E X, which are
supported on Opu,
when regarded as objects in QCohI(Opj(D )).
158
Appendices
159
160
Appendix A
How to prove conjecture 3.6.1
In this appendix we will explain how we think conjecture 3.6.1 could be proven. We
will present two different approaches. More precisely, in A.1, we will try to construct
an inverse to the functor
'I' : Ait-modiG -
QCoh (Op"').
While, in A.2 we will try to deduce conjecture 3.6.1 from the equivalence XIx
Acit-modxcG ~4 QCoh(Op"') of theorem 3.3.
A.1
First approach
Recall the BRST-reduction introduced in 2.3.1. Given a Lie* algebra 2, and a finite
set I, it defines functors
BRST1 : {U(2-Tate)-modules on X'} -+ {BRST(A')-modules on XI}
where A- id the dg-chiral algebra A' = U(2)-Tate 0 el(2). Consider now the Lie*algebra LO = g 0 Dx. As it is explained in [AG], for this Lie*-algebra, the extension
L9-T t
corresponds to the extension L.-KKil given by the Killing form -Kill
161
on .
We therefore have a collection of functors
BRST1 : {U(LKill)-modules on XI} -+ {BRST(A')-modules on XI}.
Since the critical level
given any two Ac,i-modules M
,.u is equal to -1/2rKiu,
and N, we can regard the tensor product M 9N as a U(L-KK
-nKll
-module.
This gives rise to a pairing
BRST1 : Acrit-modi 0 Acr-mod,
-4
M ON
Recall now the chiral algebra 'Der.
Vect
-+4
BRST(M
N).
A remarkable feature of this chiral algebra,
that was shown in [AG] and [FG7], is the following.
Proposition A.1.1. Let M be a chiral A,.ut-module on XI.
Consider the A,-
module (D,)1 corresponding to D,u. Then we have
BRST(M 9 ('Der)1) ~ M.
A.1.1
Construction of the inverse to T1
Consider now the functor
I1 : Acri-modiG
_+
QCoh'(Op,"I).
A generalization of the argument presented in [FG7] can be used to show the second
point of the above conjecture.
lemma A.1.1. The image of the
functor
1
restricted to the category Arit-modJG
is contained in QCoh (Op;""|).
We can therefore consider 'I1 as a functor from Acrit-modG to the category
162
QCoh
(Op"'). We now want to construct an inverse (D to I,
(DI: QCohl(Op"') -+ Acrit-mod JG
Consider the chiral algebra 'Dit, and recall that it admits two embedding 1 and r
of Acit. Denote by Ceit the chiral algebra
c,it
:= (Tx Z WxJI)('Deit) E QCoh!(Op"') 0 QCoh!(Opu"/).
Th chiral algebra Ccit naturally defines a functor
< . >: Vect -+ QCoh6(Opg",) 0 QCohl(Opg,"/),
simply by sending the unit object C to < C >*:=
cit.
The existence of the functor (D would follow from the following proposition.
Proposition A.1.2. For every finite set I there exists a pairing
< .0 . >I: QCoh'(Opey) & QCoh'( Op|j) -+ Vect,
such that, the following two properties are satisfied:
* the composition
QCoh'(Op"|) <,>,*&Id
QCoh'(Opu"|) 9 QCoh'( Opgu)
Id®<.®.>: QCoh!(
& QCohI( Opn)
Opb,)
is the identity functor.
a For M and N in Ait-mod!G, we have
< 'I'(M)
0 TI (N) >,:=< (T" z WF,)(M 0 N) >r~- BRST,(M 0 N). (A.1)
163
Let's now show how proposition A.1.2 would allow us to construct the inverse
functor GD. Recall that we denoted by B the chiral algebra
'3 = (Id M Tx)('Derit) E (Acrit-modx) Jx(G) 0 QCoh!(Op,).
Given an object T in QCohl(OpW"), we define 11 (T) to be:
1 1 (Y) :=<3I 0 T >1 E Acrit-modIG
where we regard 'B as an object in Acit-mod/G 0QCohl
A.1.1, we can immediately check that the composition
(Opu"/).
'D
Using proposition
o IF, is the identity on
Ac-it-modiG. In fact, for M a Acrit-module on XI, we have
1(IO1(M))
=< (Id Z 'FI)('Dcrit) 0 I1(M) >I~ BRST -^'ll(Dc,-it 0 M) ~ M.
Similarly, we can show that the composition xF' o D is the identity on QCoh!(Op "r).
In fact, by the first property in A.1.2, for N in QCoh!(Opu,), we have
I(G(N))=
xF (< (Id Z Px)('Derit)I o N>) .-
=<eeritOxN>= (<, >*7
A.2
< (qfx E
x)('Derit)I
@ N>=
Id) o(IdO < .O@.>)(N) ~-N.
Second approach
Let C and 'D be two abelian factorization categories. Let G : C -+ 'D be a factorization functor, G = {GI : C1
-+
'D}. The idea of the second approach is to try to
understand what it takes for G to establish an equivalence of categories over XI, if
we assume that
* G induces an equivalence Cx
-+
Dx (over one copy of the curve).
We start by the following general proposition.
164
Proposition A.2.1. Let G : C
-+
D be a factorizationfunctor between two factor-
ization categories C and 'D. If for every finite set I, G1 induces an equivalence on
Hom's and Ext1 's, then G 1 is an equivalence.
Proof. This follows from the following lemma.
Lemma A.2.1. Let G : C1
-+
C2 be an exact
functor between abelian categories.
Assume that for X, Y E C1 the maps
Home, (X, Y) -+ Home2 (G(X), G(Y)),
Ext((X, Y)
-
Ext(G(X), G(Y))
are isomorphisms. If G admits a right adjoint functor F which is conservative, then
G is an equivalence.
In fact, under the above assumptions, the functor G, admits a right adjoint F,
which is conservative. To show this, we have to show that, for every N E 'Dxi, the
functor
M - Homox, (G (M), N)
is representable, where M E Cxi.
Denote by C%0I the subcategory of compact
objects. Consider the category of pairs (X,
Morphisms between (X,
f),
f
E G,(X).
: X -+ X' , such that
#,f')= f
where X E C%1I and
f) and (X', f') are maps #
. It is easy to see that the ob ject
colimX
(Xlf)
represents the functor GI.
To show that F, is conservative, it is enough to show that for every N E Dxi, there
exists M E Cxi such that HomD,(GI(M), N) is non zero.
165
For this, consider the
exact triangle
ii*-(N)
-+
N
jj*
If i*i*(N) is non zero, then it is in the image of GI, by induction on n, since G is an
equivalence over X. Hence we are done. If ii* (N) is zero, then N is quasi-isomorphic
to
jj* (N), and we are done for the same reasons.
Proof of A.2.1. The fully faithfulness assumption on G implies that the adjunction
map induces an isomorphism between the composition FoG and the identity functor
on C1 . We have to show that the second adjunction map is also an isomorphism.
For X' E C2 let Y' and Z' be the kernel and cokernel, respectively, of the adjunction
map G o F(X')
-4
X' . Being a right adjoint functor, F is left-exact, hence we
obtain an exact sequence
0 -+ F(Y') -+ F o G o F(X') -+ F(X').
But since F(X')
-+
F o G(F(X')) is an isomorphism, we obtain that F(Y') = 0.
Since F is conservative, this implies that Y' = 0. Suppose that Z'
$
0.
Since
F(Z') = 0, there exists an ob ject Z E C1 with a non-zero map G(Z) -+ Z'.
Consider the induced extension
0 -4 G o F(X') -+ W' -+ G(Z) -+ 0.
Since G induces a bijection on Exti , this extension can be obtained from an extension
0 -+ F(X') -+ W -+Z -+ 0
in C1 . In other words, we obtain a map G(W)
-+
X', which does not factor through
G o F(X') C X', which contradicts the (G, F) adjunction.
166
0
A.2.1
The case G = 4
Now, consider the factorization functor T. By lemma A.1.1, we can regard it as a
factorization functor I : Ac,-it-mod JG -
D(QCohl(Op!nr)). By proposition A.2.1,
we have that conjecture 3.6.1 is equivalent to the following.
* For every I, the functor T, is exact.
a For every I, the functor Wi induces an isomorphism on Hom's and Ext's.
In trying to show these two points, we will use the assumption on Tx being an
equivalence. For the first point, we have the following:
Proposition A.2.2. If the functor F,, restricted to the category of strongly JGequivariant objects
T1 : Ait-mod|G
-4
D(QCoh (Opi ))
is right exact, then it is exact.
Proof. Consider the following general setting. Let C and 'D be two abelian factorization categories. Let F : D(C) -+ D(D) be a factorization functor. Assume
that
" Fx : Cx -+ D(Dx) is exact.
" For every I, F(D50(C1 ))C DO'(DI).
Then the functors T1 are also exact. This simply follows from the fact that under
these hypothesis, we also have
F1 (D 0 (C1 )) c D2:(D,).
In fact, by induction on I, for M, E D O(C,), we can consider the triangle
MI
-
j,(M'.{,)
-+
cone(MI
167
-+
Note that the second term of the above sequence is in D O(C,), and moreover, since
F1 {l,, is exact, we have that
FI(j,(M'r_1,1)) E D O('D,).
The same reasoning applies to the last term of the triangle, since the object cone(M,
j
,
-
can be written as the colimit of push-forward of modules in D20(C1 {g,),
and F commutes with both colimits and A,. Therefore we see that, by applying FI,
we get
F(M )
-4
FI(j,(M',))
-+
F1 (cone(MI
j
,
>0
EDI
which implies F 1 (M1 ) E D
0 ('D).
For the second point, we proceed as follows. If C in an abelian category, we
say that an object C E C is quasi-perfect if for any directed system of objects, the
natural map
Ext'(C, li
Ci)
-
1
Ext'(C, Ci)
is an isomorphism for every i > 0.
Assume that G : e -+ D is a continuous factorization functor between factorization categories as before. Suppose that:
" each Cx1 is generated by quasi-perfect objects.
" G induces an equivalence Cx -+ Dx (over one copy of the curve).
We have the following proposition.
Proposition A.2.3. Suppose that, in the conditions above, G 1 : Cx1
-4
Dxi pre-
serves quasi-perfect objects. Then G is an equivalence.
Proof. By proposition A.2.1, it is enough to show that G, induces equivalences on
Hom's and Ext 1 's We will deal with the case n=2, the general case can be treated
168
similarly. We need to show that for every M 1 and M 2 in
CX2,
Hom(M1, M 2 )
-+
Hom(G 2 (M1), G 2 (M 2 )) and Ext1 (M 1, M2) -+ Ext1 (G 2 (M 1 ), G 2 (M 2)) are equivalences. We can assume that M 1 is quasi-perfect.
Since A, and j,, have both left
adjoints, and F commutes with them by definition, the statement is true for M 2 of
the form j,,(M) and A. (M), for M E Cx, where we are taking the 0-module direct
image. For arbitrary M 2, we can consider the exact triangle
M2
-+
j(M'2) -+ M2
where M' is supported set-theoretically on the diagonal.
Now, such M' can be
written as a colimit M' = colimA,(Mi). By applying Hom(M1, -)
2EI
---
-
--
>.Hom(M1, M2)
2Hom(G
2 (M1),
G 2 (M 2 ))
>Hom(M1,
we get
j,(M'))
j2
>Hom(M1,
: Hom(G 2 (M 1 ), G 2 (j,(M'2)))
>
Hom(G 2 (M1), G 2 (colimA,(Mi
iEI
where the last isomorphism follows from the fact that M 1 is compact, therefore
G 2 (M 1 ) is, and Hom out of them commutes with colimits.
Note that the same
argument applies for Ext 1 's.
5
By proposition A.2.3 and proposition A.2.2 we see that conjecture 3.6.1 is equivalent to the following:
1. The functor I, is right exact.
2. The category Ait-modiG is generated by quasi-perfect objects.
3. The functor I, preserves quasi-perfectness.
Currently, we don't know how the first point can be shown.
For the second
point, for any n-duple of dominant weights A = (A,, ... , An), denotes by A},.; the
169
colimA, (Mi)),
iEI
Acit-module on X' given as
A
where ,2J
Ind
*
(V 1 9
...
-
V'
® Ox1),
(A.2)
andZ29) are the topological sheaves of Lie algebras introduced in 3.8. In
particular, for A = (0....
,
0), we recover the factorization algebra (Acit), attached
to Acrit.
We have the following proposition.
Proposition A.2.4. The category Acit-modJG is generated by the objects A.
these objects are quasi-perfect.
170
and
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